As several have noted, SLC is the best area for these kinds of trips. The Cottonwood Canyon based resorts are better for this than the Park City resorts. It looks like my favorite Motel 6 in Midvale is closed, but there are two new ones and a ton of other places in the $35-45/night range. The Alta Peruvian
is more expensive, but it's a classic throwback to the old days of skiing and worth a stay at least once in your lifetime and doable on your $1500 budget. They have dorms and regular rooms. Although bus service is fine in SLC (and best for the canyon resorts), you can usually get a car for $20/day and still save money over the fancy places at the resorts (e.g. Cliff Lodge at Snowbird). From Midvale, it's about 45 minutes to Park City. There are a couple of cheap spots maybe 10-15 minutes closer to Park City. There are dorm rooms in Park City, but not many "cheap" places for a single to stay.
There used to be a bunch of el cheapo places for singles to stay in Whistler, but that was before the place went upscale. I just did a quick search, $100 Canadien/night looks like cheap now. Ouch!
Before I started teaching, I took most of my trips this way. Many of the trips I took without any reserved ahead of time lodging. This worked especially well when I had wheels. I would roll into town and arrange lodging simply by driving around. Back in those days there were plenty of places that were not online, not bookable through central reservations and would not charge a "single supplement" (the price was for the room not per person). Because they did not have to pay referral fees and weren't high rent places, the prices were less than what you could get with package deals. Nowadays, there are enough places online that you can book a deal ahead of time. There are a lot of resort areas that have Motel 6 type lodging nearby. Just don't expect to get away with this during holiday periods at destination resorts (although I have been lucky arriving in the morning and getting a room before skiing - then finding the resort is sold out in the evening). Outside of Salt Lake, travellgin single and cheap works especially well for hitting less well known resorts.
Having wheels has allowed me to hit many different resorts on the same trip. I've done several trips where I've hit several "normally destination only" resorts in a circle around a major city. But driving in the mountains in the winter in a dinky rental car can be a bit hairy and skiing all day and then driving 4-5 hours to the next place ain't for the faint hearted either. But it sure is worth it when you can change your plans mid trip with no worries and do whatever you want with no bitching. When you "wing it" you might get burned into paying a high rate or travelling more than you planned to find a room, but you'll make up for it by adding "adventure" to your trip and being able to drive to where the snow is.
I once rolled into Targhee on Xmas eve. I stopped at what turned out to be a great Mexican restaurant in town (the parrokeet in spanish? - looks like it's history now). I asked a waitress for a recommendation for a place to stay. It turned out the owner of the restaurant also owned some cabins in town. Because he was there, I "checked in" as I paid for the meal. The cabins were cozy (e.g. instructions to leave the sink cabinets open so the pipes won't freeze). If I'd had a travelling partner, I would have been assaulted for being insane (no room and no food by 5PM on xmas eve). But when you don't have (cough) excess baggage, these things just work themselves out.
If you go in the off season, you can get some really good deals at some really nice places. I did one trip where I flew into in Spokane and took the train to Whitefish. I stayed at Grouse Mountain Lodge (one of the nicer places) for $40/night (early dec). They ran a shuttle into town and to the mountain. I then hopped the train back to Sandpoint and did a couple weeks at Schweitzer. The place I stayed at was a relative dump in town, but it was $23/night through xmas and new years and they did have a couple of cable channels and a hot tub (I think they gave me the pre holiday rate because I stayed so long).
Bed and Breakfasts can be another alternative worth checking out.