That will be so fun! I've done Mt. Rainier but not Baker. I thought this training plan was pretty good. http://www.fitclimb.com/page/mount-rainier-training-plan To the extent you can go up 14ers and do lots of hiking in advance, I think that's the best training.
Other than typical gear lists, here's what I would recommend:
Whittaker Mountaineering shop is close to Rainier and you can rent a ton of stuff - mountaineering boots, ice axe, crampons, gaiters, big jacket. Anything that you think you don't need to buy if you don't do too much mountaineering.
Clothes are pretty typical "alpine" wear -
- Softshell pant depending on temperature - I have Mountain Hardware chockstone pant with some built in gaitery thing which I use for ice climbing which I think is ideal. When I went it was actually super warm so I wore a lighter softshell pant
- Hardshell pant - the guides tell you to bring a hardshell pant, you can rent this
- Shirt / undergarment layer - pretty self explanatory, there are a bunch of brands. A lot of people like to get one with a hood to protect your neck from the sun
- Mid layer - there are a lot of brands, I've had an Arcteryx atom jacket which I've had forever, but if I were buying now I would get the Proton FL, looks really nice
- Outer layer / down - most people have an outer hardshell in the case of rain + ginormous down for when they are resting. I am a very cold person and I already had a fairly waterproof softshell that has thin down in it, which is what I wore most of the time, plus Rab Neutrino for wearing when we took a break
- Sleeping stuff - depending on what you already have and what you do outside of mountaineering. I have a 0 degree down sleeping bag and one of the new Thermarest ultralights which I think are totally worth it. Maybe you can rent this too. Or try to buy around the holidays when there are coupons
- Gloves - I have Rab women's Baltoro glove which I really like
This is not comprehensive, there are a ton of detailed trip reports you can look at where people talk about what they brought. All of this stuff you can check out REI outlet, I personally like Mountain Hardwear, Outdoor Research, and Rab as reasonably priced brands. Sometimes Mountain Hardwear and Rab have nice sales on their sites, and I think Arcteryx does now too if you want something fancy.
Have fun, train, and let me know if you have other questions!