Rainier is the classic test of North American Mountaineering. You need to do it before you go any further.
If you've never done it before, there are really only two standard routes. Disappointment Cleaver and Emmons Glacier. Emmons is longer, with more vertical on summit day, but not quite as steep. It's also a bit more remote and offers a better wilderness experience, since the Park Service keeps the number of overnight campers along the route fewer than along the DC route.
The DC route, being more popular (a few thousand people a year try it) is also better marked and maintained, often with all the crevasses and bridges over them very well wanded (the little 3' flags), and in those few spots where you go along or over rock there are sometimes fixed ropes. Sometimes even in a whiteout you can just stay in the trench that hundreds of feet made along the path in the previous 24 hours.
There are a few other seasonally "easy" routes, such as Gib Ledges and Ingraham Direct, both of which are in condition early in the season - April or May even, and then as quickly out of condition. If you are an experienced Ice or Steep Snow climber, the Kautz or Fuhrer Finger routes can be fun in May or June.
The route you select determines a lot. Emmons is entered from the White River entrance, and a long but pretty hike with not a great deal of serious steep vertical leads to Camp Shurman, at 9,500', where you'll find a relatively nice toilet, barrels to dump your blue bags (on Rainier, if you aren't using an authorized toilet, you have to do your more solid business into a blue plastic bag and haul it with you - it's not as hard as it sounds) and about 30 tent spaces relatively sheltered from the wind. From here on summit day, you'll ascend through the crevasses along a path called "The Corridor", pass the bergschrund (the large crevasse at the top of the glacier - sometimes amazingly deep and wide and far across with the narrowest and winding bridge across of firm snow) at around 13,000', then angle back to the summit, Columbia Crest, at 14,411'.
The DC route leaves from the Paradise Lodge parking lot at 5,500' and goes up to Camp Muir at 10,000', a steep climb in a short distance. From Camp Muir, on summit day, you climb up through Cathedral Gap to the Ingraham Glacier, over to Disappointment Cleaver (a large and crappy rock feature). Depending on the time of year, you would go up along the cleaver, or along one side and up to the Emmons Glacier for the hike to the volcano crater rim, where you finish by hiking about a half mile to the summit.
If you're going to camp anywhere on the mountain, you'll need a wilderness camping permit (specifying dates and campsites), available online, in the mail, or from a couple of the ranger stations. If you're going above either Muir or Shurman, you'll need a climbing permit (unlimited use for the year in which you buy it) available as above. Guiding companies normally take care of this for you (so it's only significant if you're not using one).
Important skills to have include walking comfortably in crampons. In the dark. On ice and snow. On crappy gravel mud and rock. If you're in really good shape and have good balance and coordination you can figure this out on the fly, but I've seen V4 climbers break down in tears at walking on 4th class terrain in crampons. Rope management is essential - part of the safety system is to keep the rope a certain level of tight. This keeps your falls below a Fall Factor 1. Way below. You have to walk the right speed to do this, while watching your footing, while keeping your ice axe in the right hand, pointing the right way, while breathing thin air. In the dark. Uphill in a headwind. Yeah.
Self-arrest is important as well. The ability to stop your own slide from a number of angles is helpful. Stopping a ropemate's fall or slide is normally done only one direction and on purpose - not as tough to learn.
Climbing a rope using a prusik knot ascension system is helpful, though with a guided group there will always be a few handy guides to haul you out. The myth is that you can't go anywhere on Rainier where you can throw a dead cat without hitting a guide. Keep that in mind - always bring a dead cat in case you fall into a crevasse.
Most guided groups will take time out either before or during the basecamp stay to give a quick course in these skills, but it's probably a good idea to pick up whatever you can however you can beforehand - you can't know too much unless you try to "share" it with someone who doesn't really want to know how smart you are.
As far as equipment, Wal-Mart is pretty much out. I've had some sad team-mates before who tried that route with potentially life-threatening results. You'll need special clothes, special boots, special equipment. Most of this stuff can be rented in Ashford Washington, where two of the three guide companies authorized to guide on Rainier keep their headquarters. Otherwise several good stores in Seattle rent as well.
If you balk at the price of the boots and gloves, remember to divide it by ten for your "dollars per toe" figure to compare it to the long-term cost of amputation. If you go to the guide company sites you'll find really good equipment lists based on the average climber. If you really know yourself at altitude in the cold, in a storm, then do your own thing, otherwise, sticking to the lists is the safest course of action.
If you're going with one of the guided groups, that's about it. If you're going up on your own (in which case you're obviously wasting your time reading this since it's way too basic for someone as cool as you) or with a friend or whatever, besides the above gear, you'll also need to consider food, tents, and cooking equipment and fuel, since the guide companies normally take care of that.
Anyway, that's a good first thought logistics guide for Rainier.