Hej, I lived in the U.S. before moving to Sweden and skied in Colorado frequently so I can give a few tips. Some of the answers to your questions depend on where exactly in Colorado you are going and how much skiing you want to do.
Unfortunately, it is probably not going to be cheap. If you look at "ticket window" (meaning show up to the ticket office and buy a ticket on the day you want to ski) single day lift ticket prices in Colorado you will be shocked. Car rental also will be expensive.
If you want to ski many days, then it likely is cheapest to buy a multi-day or season pass of some type. The "Epic Pass" and the "Ikon Pass" are two of the most popular options. They each come in different forms, ranging from a few days to the entire ski season. You should read about each one (and their many variations) carefully to decide what best fits your plans. Whichever you choose, it is cheaper to buy the pass online early, before mid-November usually. Typically, you will pick up the pass at the ski resort.
For lodging, you may want to start looking at AirBnB, which usually has many options. Things will begin to get booked up as the season nears, so I would start looking early.
My favorite place to ski in Colorado is Arapahoe Basin, which is a smaller but excellent independent resort. The big names - Vail, Breckenridge, Aspen, etc. are nice, big, crowded and expensive. The smaller names - Arapahoe Basin, Loveland, Copper, Winter Park have comparable skiing and are more chill and less crowded.
If I was going to be in Colorado for 3 weeks and wanted to ski as much as possible as cheaply as possible as easily as possible, I would buy an Araphoe Basin "mid-week pass" or a "Keystone plus pass" and rent an Airbnb in either Silverton or Frisco (nearby towns). I would then ski Arapahoe or Keystone on repeat Monday through Friday. There is a free bus service within Summit County called Summit Stage that could be useful for you if you want to try to pull the trip off without a car. If I got a "Keystone plus pass" I would spend 2 weeks near Keystone and 1 week at Crested Butte.
If the goal is to travel all around Colorado trying many places, then you'll have to string that together for yourself but I would still buy some kind of pass early and look at Airbnb early. If you're only going to ski a few days during the 3 week trip then much of what I have said doesn't matter that much.
Trevlig resa! Colorado is a great place.
