Normally, you would have special equipment, either purchased or rented just for this climb. If you know yourself, and your equipment needs, and have committed to a future of doing mountaineering, considered by many to be merely insane or even socially detrimental, then purchase is a good option with many long term benefits. If you have any doubt at all, then you really should rent.
You can rent almost every single last bit of gear and clothing, including long underwear at one of the many REI's in the greater Seattle area (if you go to their website, store locator, and click on the little map icon for Washington, you can find their stores, then go from there to the Rent Gear page where you can get a list of items available at each store - the list sucks a bit, Mountaineering Gear is as close as you can get, and I never called one to find out what that means). Feathered Friends and Second Ascents may be good options as well. Alpine Ascents has their office in Seattle, but International Mountain Guides and Rainier Mountaineering Inc. have stores and rental offices in Ashford (the last small town before Paradise in Rainier National Park).
If you are a backcountry skier you probably have much of the required gear, and in fact, you might be able to make your AT boots work. I've seen many guides climb it in Scarpa Denali's. If you are a serious skier you might have some of the basics. If you aren't so serious about winter sports (or your idea of winter sports includes firearms and/or small gas motors), you'll probably need a lot of stuff.
|From Liberty Ridge Rainier 2009|
Starting from the inside and working out:
Undies: some people wear them and some don't - underwear by Patagonia, REI, Smartwool seem to be the best overall. Some of the sports underwear are meant for moisture transport while running, which you won't be doing. Those will get soggy and you'll lose heat too fast. Same for cotton, heavy wool, wool/poly blends, multi-layered, etc. Don't do it. You'll need one. You won't be changing.
Bra: IF you need one, again, be cautious of those that trap moisture because they're designed to have a constant 10 mph breeze as you run in them. One. No changing.
Base Layer: fancy word for Long Underwear. This is the root of your clothing system. Most manufacturers make three variations. Thin, Medium, Thick, sometimes with fancy names like Cap2, or Expedition, or 260g or whatever. Simple enough to figure out. You will need one for the top, and one for the bottom. I myself think even the thinnest wool from Smartwool is too thick and traps too much moisture for a multiday trip, but ymmv. Wool smells better overall after several days, but some of the others aren't too bad. Unfortunately they vary depending on your body chemistry and you don't know till you try. You won't be changing these either. Some people like turtlenecks, and some like zip mock turtles. Make sure the fly is useful for you - some are skimpy or in odd places, depending on your anatomy. If you're a girl who will be using a pee funnel, you might have to use men's bottoms.
Mid-layer Top: A second base layer, though it's not a base layer if something is under it? A mid-weight baselayer, expedition weight baselayer or very thin softshell or wind shirt are all common for this application. If you don't know, you may be happiest in a PolarPro fleece designed for running. My personal preference used to be a wind shirt but I've been using The North Face Apex Bionic softshell for almost a year now. It's thin, light, zips for ventilation, is about 80% wind and waterproof, and breathes better than most fleece imho. In mild weather I can wear it for my outer layer.
Softshell Top: Your main outerwear top. A mid-weight softshell like the Marmot Sharppoint, Mountain Hardwear Alchemy, or Arcteryx Gamma series. Abrasion resistance, breatheability, water resistance, wind resistance, and warmth are all features to keep in mind. The North Face and a few others also make good choices. Go try some on. Remember, you will be wearing this over everything above probably all the time, so make sure it is big enough and that you can move freely.
Softshell Bottom: Abrasion resistance, freedom of movement, warmth, water and wind resistance. You will be wearing this the whole time. I have a pair of Cloudveil ice climbing pants I wear a lot, a pair of Mountain Hardwear Syncro I wear sometimes, and a pair of TNF Apex Bionic Bibs (no longer made) that I absolutely love. Make sure the fly works for you with all the above on (and with a harness on if you can) and if you're a girl using a funnel. Some features that are love it or hate it, are moon drop seats, zippers that go from front to back, various ankle closures that can take the place of gaiters (if you don't know I won't be teaching you now - stick to gaiters if you're a beginner). Ski pants suck - the ankles are too big and you'll trip over them with your crampons. Crampon patches are normal on pants meant for this use - you WILL poke holes in your pants. It's a war wound. It shows you really do walk around in crampons. Yes, you paid $400 for your pants. Oh, well. Duct Tape.
Goretex Pants: This isn't the place for experimenting if you're a beginner. Goretex is the standard (but completely equivalent membranes can work too so long as they are as durable). It needs to go over everything above. You will most likely keep it in your pack the whole time and only put it on when the wind picks up or it starts snowing or raining bad. Hence, you will need to be able to put them on, with a harness, with crampons, in a windstorm/blizzard. Full length or 80% side zippers help. Make sure you can actually put them on in the store. I've seen lots of people show up and not be able to put their own pants on because they never did it and don't know the trick of lining up the zippers.
Goretex Jacket: See goretex note above. It needs to go over everything above. Some people also like them to go over their parka, see below. Keep that in mind when sizing. Since goretex is neither waterproof nor breathable (that's a little bit of a joke I picked up in Alaska) you want to make sure that your pit zips are good enough, and that you can operate them in a blizzard, etc. with mittens on. I have a jacket that has a hood that zips up into the collar. I should keep it there because it's annoying to me anyway. You probably don't want that, so make sure the hood is sufficient to cover your helmet and provide visibility. You'd be surprised at the expedition goretex out there that doesn't work right with helmets. Same for "Ski Jackets". You don't want a lot of extra crap, like pockets - you won't be using any pockets in a blizzard. Trust me. You also hope it never comes out of your pack.
Puffy Jacket: Your main survival outer layer. Down or synthetic? How thick? For Rainier, most people can make do with whatever, your life doesn't really depend on it like in Alaska. Patagonia DAS Parka, OR Chaos are great synthetic. The standard down jacket is the Mountain Hardwear Subzero Parka, but I think I'd rather have the Mont Bell down parka - better baffles and down. Lighter. You will pop this on and off at every rest, at camp, and as you near the summit. It will get a lot of wear and tear, wadding it up in your pack.
Gloves: Get a system unless you know yourself well enough. OR, MH, Black Diamond, all make good systems. A system is a liner and shell integrated so you can use any combination to make three separate gloves. Wear the liner in camp or when it's warm. Wear the outer when it's colder or windier and maybe wetter. Wear them both when it's colder and windier still. Most of these come with "moron straps" use them correctly. Ask someone (but not at an REI, they most likely don't know) how to use them to prevent glove loss. Some people like an extra liner pair - you can get something a bit thicker, or with a patch of rubber for better climbing ability, and make sure they also fit the outer shell if you like. You will most likely be doing a lot of equipment things with your hands, so a slippery palm/finger area sucks.
Mittens: The gear lists suggest a pair of very big and heavy mittens on the summit. I've never used them. Most people I know have never used them. But then divide the weight and price by 10 fingers. I've used heat packs before but didn't like them. Maybe I've never been high enough or cold enough when I've had them. Your call, but if it's your first time, do it just to be safe.
Socks: Sure, wear socks. What most experienced people I've talked to that were in love with their boots do is buy boots that fit. Then fill them with socks. Simple. But for the rest of us/you, get the thick expedition weight socks. Pile or loop, wool or wool blend, but not the ragg wool. Ragg wool will never dry out in your sleeping bag at night. Acrylic hunting socks don't work good either. Most ski socks will be too thin, and since they aren't meant to be walked in, might bunch up from friction and cause pain. You probably want to have at least two pair, so you can be drying one with body heat while the other is on your feet. Some people have a "sacred summit socks" pair that they only put on for summit day. For a three day trip that seems excessive.
Hats: Generally, a wool or wool blend hat that fits under your helmet is best. Some lists put in a separate balaclava. Synthetic balaclavas work good, but like socks, ragg wool is out. Some people bring just the hat and then a buff to double as the balaclava, but they're often not as wind resistant. I prefer a thinner lighter polyester hat and a buff, but that's just me. YMMV.
That's it for clothing. Really.