For finding a cheap place to live, I would think Salt Lake City is going to be hard to beat. You might look at VRBO
to see what is out there. I'd also check local listings on Craigslist
for the Salt Lake and other areas you are interested in. The Salt Lake City area is a large metro area with super quick access to the mountains. Lot's of places you can rent from that are not necessarily ski oriented at all. Prices should be better if you can find a place to suite your needs. Look at stuff on the East side of town first. The further West you go the longer you are from the mountains.
For passes, it is kind of hard to beat the bang for your buck you get with Colorado passes. The Epic Pass
gets you most of Summit County, plus Vail, Beaver Creek, and Heavenly and Northstar in Tahoe. Several resorts on one pass for $669. Prices will go up. So buy soon if you are going to do the Colorado thing. Be warned though, the savings (which are big) on the pass might be awfully hard to make up due to the cost of finding a place to stay. Denver in all reality is too far for a reasonable base of operations. At the very least you would want to be in Idaho Springs, but Summit County proper would be the place to stay. Idaho Springs is a bit of a mountain kook town. Alright for us Denverites to pass through, it wouldn't be all that fun to stay there for weeks.
Tahoe would be another good choice. Has much of the same problems as Colorado. You could try for Reno for a cheap place to stay. The drive to the hill from there is more than Salt Lake city but relatively easy to spots like Mt Rose. Again, it's a city that isn't just reliant on ski tourism like Summit County.
For a car, I would say just buy a cheapy. Car rental prices are going to cost more than spending $500-$1k on a beater vehicle you can drive the hell out of and push off a cliff when you are done with it. Or you could try to sell it for about what you bought it for before you leave. Had some Frenchies pull this off on a climbing trip they did out here 5 years ago. You should be able to find some vehicle from the 80's that'll fit the bill. It's not like you'll be here long enough to worry about emission testing for it. Not sure about insurance, but you should probably try to have a liability coverage on it, just so that if get in a wreck, you're not in hot soup.
For flights, Southwest
is having a big sale for continental flights in the states. It ends today unfortunately. They are one of the more reasonably priced airlines though, and they have no bag fees for the first two pieces of checked luggage. Including your board bag. Overall one of the best airlines to go with if you are a budget minded boarder.