Who lives in Seattle is probably used to seeing the snowy mountains of the Olympic Range on the other side of the Puget Sound. For me, that's still one of the great views of this area. Every time you see these mountains, it seems almost as though they are calling you, inviting you to cross the Sound and explore the other side. This area is the Olympic Peninsula, home of one of the most unique National Parks of the United States: Olympic NP.
This park includes three main sections. Hurricane Ridge near Port Angeles, is the primary access to the mountain area. The Hoh Rain Forest, near Forks, is the famous rain forest. Then there is the cost, separated from the rest of the park, but part of the park itself. The beaches are spread all along the coast, but the main areas are Kalaloch, La Push, and, further north, Ozette where you can hike to the famous Shi Shi Beach with its Point of the Arches.
Olympic NP is huge (about 3,700 square Km), and covers pretty much the whole Olympic Peninsula. Don't even think you can visit this park in a few hours, it's just not enough. Moreover the distance between the park's different areas are significant, especially considering that some locations can only be reached via low speed backroads.
That said, I would suggest at least three days to enjoy the whole park. That should be enough to explore a bit of each area and enjoy the unique landscape without rushing.

Introduction

All areas of the park can be reached via rather short detours from Hwy 101, which follows the boundaries of Olympic NP along almost the entire peninsula.
In most cases these detours are pretty short mileage wise but could take significant time due to the low speed limits.
For those of you starting from Seattle, the fastest way to reach the park involves crossing the Puget Sound by ferry. The short trip is about half an hour long, costs a bit more than $10 per vehicle (plus a per person fee) and allows you to reach the northern area of the park in a couple of hours. You can get a ferry in downtown Seattle at Pier 52, Bainbridge Island destination.
As I said, there are two main choices in terms of accommodations and, if you want to visit both the northern and western part of the park, my suggestion is to plan two stops. Port Angeles is the best choice for the northern area, a few miles from Hurricane Ridge. This area open most of the year but the road might be covered in snow as late as July. That is also the best season to visit the Hurricane Ridge mountain, when flowers are blooming and all trails are snow free.
Forks is the largest town on the western side of the park, with plenty of motels and restaurants and not far from several pf the park's beaches and the Hoh Rain Forest. Here I recommend the Forks Motel, where for a moderate price you will enjoy comfortable, clean and welcoming rooms. I also enjoyed dinner at The In Place, on the other side of the street right in front of the Forks Motel. Their huge hamburgers go for less than $6. The Smoke House Restaurant Is also a good choice for fresh sea food, but expect to pay in the neighborhood of $20 per person.

Hurricane Ridge, Lake Crescent & Sol Duc

Living in a city with the reputation of being “the rainy city" I had to quickly learn how to deal with rain. When it comes to outdoors, the easiest way to go about it, it's just to know which times of the year are the best to minimize your risk of rain. Mountain parks live a very short “tourist" season and it's extremely important to be timely, so that you can enjoy that brief period when snow has melt, and flowers bloom and cover meadows. If you go to Hurricane Ridge (17 miles away from Port Angeles) in August, blooming is already past its peak and just a few mountains are still snow covered. Late June or early July, depending on the year, it's probably your best bet. Whatever is the month you choose, the view from Hurricane Ridge is amazing. Clouds are part of the landscape and often cover many of the peaks making the scenery even more dramatic and enchanting. From the top you can enjoy not only spectacular views of the surrounding mountains, including Mt Olympus, but also an extraordinary view Port Angeles' harbor and the San Juan Islands.
Immediately around the visitor center many short trails lead to the meadows and several viewpoint. The trails are all fairly short and intersecting frequently. Walking for a couple of miles you'll see several examples of what this area has to offer. If the season is right, do not miss the blooming. It's not going to be as spectacular as in Mt Rainier, but is still a beautiful experience.

The detour to Sol Duc takes away a good 40 minutes one way but if you're not in a hurry you'll be rewarded by a very particular and fascinating area of the park. Access to Sol Duc is in the northern part of Olympic NP. The road is heading south getting very close to the Hoh Rain Forest making this area unique in that you'll see a fusion of elements from both the rain forest and the mountain area. Sol Duc is famous for its hot springs, which you can access for a fee. The main naturalistic attraction is Sol Duc Falls, which can be reached by a 0.7 miles one-way trail. These waterfalls offer a great show and are certainly worth the detour from Hwy 101 and the short hike required to reach them. The trail passes through a forest where you can also see elements of rainforest. Overall an excellent trail.

Lake Crescent is a beautiful lake surrounded by forests and snow covered mountains. There are several opportunities for hiking. The most famous trail is the Marymere Falls Trail which leads to the homonyms waterfalls. In the first part you'll be getting a the first taste of the hanging green moss that you'll seen later in much larger quantities in the Hoh Rain Forest. The trail is not particularly demanding and the falls are worth the visit.

Hoh Rain Forest and Lake Quinault

The only rainforest north of the equator is one of the great features of Olympic NP and is accessible by tourists in several areas of the park. The Quinault Rain Forest, in south-western area of the park, and the Hoh Rain Forest, a few miles from Forks.
Driving to the Hoh Forest is a pleasure in itself, following the beautiful Upper Hoh Road that slowly winds thought the forest passing under huge trees and hanging moss.
This is also one of the best areas for watching wildlife, including the famous Roosevelt Elk. Always keep your eyes open! There aren't a lot of short hikes in this area but there two best are the Spruce Nature Trail (1.2 miles) and the Hall of Mosses Trail (0.8 miles). Both take you che mostrano la foresta nelle sue diverse forme. At very close contact with the forest. They are both very simple, with virtually no elevation gain, and lots of interpretive signs that will give some background information on everything there is to know about the rainforest.
The best way to enjoy these trails is by simply walking and looking around. What I suggest is to not only focus your attention in front of you and on the sides of the trail. There is a whole world above and below you, with plants that grow on other plants, with the famous (and disgusting!) Banana Slugs crawling and branches from which hang "beards" made of moss, which in some places become so dense to block almost completely any sunlight.
One more interesting and (much) longer trail is the Hoh River Trail (17 miles). This is a backpacking adventure, which leads up to Mt Olympus. The Visitor Center is very small and when I went, although open, there was no ranger. So try not to get to the Hoh Rain Forest completely unaware of what will expect you, you might not find a ranger to answer your questions.
Further south is Lake Quinault, surrounded by the same rainforest that the majority of tourists sees in the Hoh Rain Forest.
In this area I recommend the Maple Glade Nature Trail, a rather short but really beautiful trail (in my opinion, better than any of the short trails in the Hoh Rain Forest). The two sides of the lake (north and south) are accessible via two different routes (North Shore Road and South Shore Road) which form a loop around the lake. Sia la parte nord che sud hanno accesso gratuito, ma nella parte sud in alcuni parcheggi è richiesto un Forest Pass. Both the north and south ends have free access, but on the south end a Forest Pass required to park in some parking lots. If you have a National Park Pass you do not need the Forest Pass.
There are many other sights such as the World's Largest Sitka Spruce (south end), the Rain Forest Nature Trail (south end, pass required for parking) or the Big Cedar Tree Trail (north end).

The Coast

Olympic NP incorporates much of the coast of the Peninsula. There are essentially three different areas, not directly connected and quite distant from one another in terms of milage.
To the south is the area of Kolaloch that includes a number of beaches numbered (and named) 1 to 6 , plus Ruby Beach. The beaches are sufficiently similar to allow you to choose one or two without worrying to miss much from the other ones. Beach 4 offers perhaps the best opportunity to see the famous tide pools, a kind of natural pool formed among the rocks, where starfish, crabs and other marine wildlife gets stuck at low tide. Ruby Beach is perhaps the beach I would recommend if you had time for only one walk in the coastal area of Olympic. Here you can check out both the sea stacks and the tide pools. E' abbastanza scontanto ma è bene sottolineare che per godere delle tide pools dovreste esplorare la spiaggia mentre il mare si ritira verso la bassa marea. It's pretty obvious, but worth remembering that the best time to check the tide pools is at low tide.
North of Forks is lies a second coastal area. After leaving Hwy 110, the road bisects,  leading respectively to Mora with its famous Rialto Beach and La Push, with its 1st Beach, 2nd Beach and 3rd Beach.
Rialto Beach is a beach covered in stones and super smooth broken logs. Both south and north of the beach parking lot you'll enjoy great views of sea stacks, often semi-hidden by low clouds and fog. A trail runs north, and a 1.5 miles hike will take you to the famous Hole in the Wall sea stack. I enjoyed Rialto a lot. The beautiful 360 degrees views include sea stacks everywhere, huge broken logs, fishermen houses and the forests that surrounds the beach to the east. Whether it's foggy or sunny, Rialto will not disappoint you. When it's foggy you'll have a déjà vu from pirates movies and if it's sunny you'll have a chance to enjoy a sunset like no other. Moreover, unlike other beaches in the area, parking is a few meters from the beach and no hike is required to enjoy the various sights (unless you want to get under the sea stacks).
If you want to combine the coastal views with a nice hike, then 3rd Beach is a must. This hike is approximately 1.4 miles long and passes through an enchanted forest. Try to take this walk in the morning and I assure you that the fog among the tall trees will prompt you several times to wonder if you are awake or not. Once you reach the beach there is still a little walk if you want to get close to the sea stacks. Along the way, you can also see a waterfall flowing from the top of a gully directly into the sea, while waves are breaking on the rocks just below. If you want a little hike to "earn" your access to the beach, I cannot think of a better hike in this area.

Just 16 miles to the north (but about 55 miles and one and a half hours by car) you'll find Lake Ozette. Here I strongly recommend Triangle Trail, which leads to Cape Alava and Sand Point. If you want a good backpack, then you will not find much better than Cape Alava. The campsite is absolutely fantastic!
If you want a even longer backpacking trip, then plan another day to get to   Shi Shi Beach and its famous Point of the Arches. For those who do not want to backpack there is another way to see Shi Shi Bach. This beach is accessible from the north, from the Makaha Indian reservation. Below is a description of this hike.

Shi Shi Beach and Point of the Arches (10 miles, 3 1/2-5h, round trip, almost flat)
First thing, you need to find out where the trailhead is, and this is already not an easy task. As I said, the trailhead is within the Makaha Indian Reservation, outside of Olympic NP boundaries. Lasciata la Hwy 101 poco dopo Lake Pleasant imboccate la Hwy 113 north, poi la Hwy 112 north e raggiungete Neah Bay . Leaving Hwy 101 shortly after Lake Pleasant take Hwy 113 north, then Hwy 112 north and drive to Neah Bay. Here stop at Makaha Museum. Whether you want to learn  more about the local history and culture or not you'll need to stop at the Museum. This is the only place to get the parking permit you'll have to display at the trailhead (technically they call it a Recreational Use Permit ). It only cost a few bucks and is good throughout the calendar year. You'll also get a detailed map of the area. Follow the directions and in about 30 minutes you will get to the trailhead.
The first part of the trail offers not much more than a 2 miles flat hike in mud, with Banana Slugs here and there and a dense forest covering any views on the beach. When you'll begin to ask yourself if you missed a turn and if you're still on the right trail a very steep and short descent will finally get you down to the beach.
You made to Shi Shi Beach! Qui finalmente gli sforzi sono premiati anche se ci sono ancora circa 3 miglia da fare per arrivare al Point of the Arches, già comunque visibile in lontananza. Here your efforts is finally rewarded even though there are still about 3 miles to go to get to Point of the Arches (already visible far to the north). The area is very wild and often you can enjoy the beach in almost complete solitude. Here, the sea crashes against the sea stacks with unimaginable power. The centuries long battle between the sea and the sea stacks is one worth the long hike. Some of the sea stacks take strange shapes, bowing and opening themselves almost like in a bullfight in which the sea is the bull and the sea stacks are the bullfighter. This is the kind of trail that I call "contemplative" since there is not much else to do than walking contemplating the landscape around you.
Once you get to Point of the Arches, check your tides table (which you should have already done much earlier) and sit back to enjoy this furious battle between ocean and earth, for as long as you can. Just keep an eye on the tides!

Tides

If you want to hike on the beach, you must be aware of tides. There are many people who have been even killed by being in the wrong place at high tide time. At the Visitor Center in Olympic NP, ask a ranger for a tides schedule. The table is very clear. Read it carefully. Every day the tide schedule change a lot, so do not try to guess tomorrow's tides based on today's. A tide schedule is very easy to read. For every day you'll find numbers like in this example: 4.0L 2:04. It tells you that at 2:04 am there will be a low tide (L) which will cover about 4.0 feet of beach. So, how do you use this info to plan your hike? The idea is that the sea submerges the beach at high tide and exposes the beach at low tide. You could start your hike when the sea is retreating, leaving space to walk on the beach and try to turn back shortly after low tide, before the sea starts to submerge the beach. If you'll remember only one thing from all of this, this is what I want you to remember: be careful!

Cape Flattery

I must say that I had prejudices against this place, mainly because I thought that a place that is known as "the most north-western place in the continental U.S." was to be more of a tourist attraction like Four Corners than anything else. It was with great pleasure that I was proven wrong. From Neah Bay, you need to drive for half an hour on a nice paved road. Then a short trail one mile trail leads to several viewpoints on the Ocean, all of which are truly amazing. Here the wind is blowing strong, the waves are breaking on the sea stacks and crashing huge logs as though they were small branches. These views are of wild beauty and, if you are lucky, and the time is right, you could have the chance to spot whales.