For people like me who grew up in a city like Rome and are not used to mountains, the view of Seattle's skyscrapers framed by the Cascade Range snowy peaks is one I get never tired of.
When Summer comes many of these peaks' white hood melts off and at by the end of July only one of them is still covered in snow: Mt Rainier. This giant is 4392 meters high, so massive that, in cloudy days, you can still figure out where it is by looking at the strange and unnatural shapes that clouds are taking in order to get around the mountain. Mt Rainier lives at its own pace: it lets the snow accumulate during winter and it wakes up lazily in late Summer. But its awakening is a show you do not want to miss. The lucky ones who can enjoy a crystal clear sky, snow covered fields and hundreds of flowers of all colors will not forget this show for a long time.

Introduction

Mt Rainier is located southeast of Seattle, WA about a two hours drive by car. The park consists of several areas but the two most famous are certainly Paradise in the southwestern corner, and Sunrise in the northeastern corner. Other less visited areas included Longmire (just after the southwestern entrance), Ohanapecosh (just after the southeastern entrance) and Carbon River (near the northwestern entrance). You can enter the park from north east, driving on Hwy 410 and this is definitely the best choice if you're headed to Sunrise.
The primary access to the park though is Hwy 706 that leads to Longmire, Paradise and then continues up towards Sunrise becoming Hwy 410.
From Seattle you can visit Mt Rainier in one day, spending the morning in Sunrise and the afternoon in Paradise. My advice however, is to plan a couple of days in the park. This will give you enough time to enjoy Mt Rainier without rushing. In this case you could sleep in one of the two lodges inside the park, the National Park Inn in Longmire or the Paradise Inn in  Paradise. Just outside of the park there are several lodges, almost all of which are concentrated around the southwestern entrance where you'll get not much more than the usual combination of motels, a general store and gas station. To sum it up, if you have only one day available, I'd recommend you spend it all in Paradise. With a little more time, say a second day, you can visit Sunrise and also enjoy several of the attractions along the road.

Hiking

Mt Rainier is a paradise for hikers but is much more than that. Even on a cloudless Summer day the heat will always be tolerable and the mountain fresh air will make hiking a lot easier than in the southwest heat. Don't forget your sunscreen though! As you enter the park you could give a quick look at the Trail of the Shadows (rather interesting), an easy interpretive trail where you can lear about Mt Rainier's flora and fauna. Dozens of trails start from around Paradise's visitor center. The trails intersect often, and you can choose any combination of two, three or more trails to explore the area. For example, when I visited Mt Rainier for the first time I followed the Alta Vista Trail to its intersection with the Skyline Trail, I continued up to Panorama Point and I came back via Dead Horse Creek Trail. Ask the Rangers about the trails' condition. Many trails may still be partially covered with snow even in Summer, and some parts may be damaged after severe winter weather (it's not rare that some bridges have been destroyed by the Spring's flooding).

Alta Vista Summit, Skyline and Dead Horse Creek combination (6 miles, 4-5 h, moderate/strenuous, loop, 1700 ft elevation gain)
While you approach Mt Rainier by car the huge mountain seems almost as though it's daring you to conquer its top. It's almost difficult to control the desire to start hiking and prove Mt Rainier you're worth the challenge. Many trails starting from Paradise can help your thirst for hiking. Paradise is famous for its meadows covered in flowers, easily accessible from almost any trail you choose. The most famous of this is perhaps the Alta Vista Summit Trail and that is an excellent first hike in this area. The trail climbs gradually passing through green meadows covered with flowers of every color. What is most striking is perhaps the number of flowers that virtually cover every inch of space. A fresh sent fills the air. Remember though that the flora is very delicate and all the trails are full of signs that warn of not stepping off the trail.

If you want to get higher and a little further away from the crowds leave the Alta Vista Trail and continue on the Skyline Trail. The goal here is to reach Panorama Point. This part of the trail is not paved and even in summer we had to deal with sections covered with snow. It's amazing to see how as soon the snow melts there are already plenty of flowers ready to bloom. Going up we met a couple of marmots, apparently undisturbed by our presence while eating flowers and running on the snow. The stretches covered with snow are pretty slippery but walking in the snow while the summer sun is shining undoubtedly a nice experience. While you keep climbing the views of the surrounding mountains become more and more spectacular, with Mt St Helens, Mt Adams and snow capped mountains, and glaciers all around you. Once you get to Panorama Point you'll enjoy one of the finest views in this area, a 360 degree show that reminds a little of a mountain version of Utah's Muley Point.
Here you can stop rest a little and enjoy a sandwich, because whether you're a desert fanatic or a mountaineer these are views you do not see often. On the way back you shouldn't miss the detour to the Glacier Vista, a viewpoint from which you can the Nisqually Glacier. The loop the Dead Horse Creek Trail, which offers beautiful views of Mt Rainier, towering mountains and wildflower meadows.
The loop is long and quite strenuous but definitely one of the best hikes in Mt Rainier.

Comet Falls Trail (3.8 miles, 3h, strenuous, round trip, 1600 ft elevation gain)
We chose this pretty much only to see one of the most spectacular waterfalls in the northwest and, to be fair, the Comet Falls truly deserve that title. The trail passes through a very dense forest and then follow the course of the Van Trump Creek while you are climbing a very steep and exposed slope. The trail itself is nothing special. But the effort required is more than offset by the final view of Comet Falls. These falls are about 100 meters high and really majestic, so powerful that they can fill the air with a mist up to a hundred meters from the waterfall.

Shadow Lake (3 miles, 1.5h, moderate, loop, 600 ft elevation gain)
In the Sunrise area any trail you choose will provide magnificent views of Mt Rainier. The Shadow Lake Trail is one of the moderate trails starting just outside of the visitor center. It climbs gradually following a rather exposed trail. Small sections surrounded by trees are alternated by alpine meadows. Mt Rainier though is perhaps the only real attraction of this trail. Seeing the mountain so close is really impressive. Shadow Lake itself is a relatively small pond I personally have not enjoyed too much. After having stopped a couple of minutes, just move on. From here on, the trail is virtually downhill and, unfortunately, Mt Rainier will be behind you. All in all this is a trail that I would not recommend too warmly, considering that the view of Mt Rainier is virtually the same you'd enjoy from the parking lot. The only exception is during blooming or Fall when colors are at their best.

Along the Way

Comet Falls are truly spectacular, but all in all Mt Rainier offers other beautiful waterfalls much more easily accessible, pretty much a few meters from the road. Christine Falls can be seen a few meters away from the Comet Falls trailhead and the beautiful Narada Falls can be accessed by a short 0.1 miles trail.
Once you leave Paradise spend a few minutes stopping at Reflection Lakes and Lake Louise, two small lakes which reflects Mt Rainier and offer an excellent opportunity for a picnic.
Continuing east you'll get to Steven Canyon, where I'd recommend you to check out Box Canyon. Here you can admire one of the deepest and narrowest canyons of this region.
In the Ohanapecosh area, I'd highly recommend the Grove of the Patriarchs Trail. The trail is not long (1.3 miles) and will take you through a Douglas fir forest with some of the tallest and biggest tree in Mt Rainier. Children will especially enjoy this hike.