Norquay (small, steep, not a lot of snow but good park and night riding wed & sat) is right in town, visible from the streets, and although that close the sketchy access road make the trip up about 10-15 mins.
Sunshine Village is 18km away down the trans canada, and generally takes 20-25 mins from Banff by car / bus.
For some reviews of the three, check out snow-forecast.com by searching for each resort and clicking on reviews.
Next closest is Lake Louise, about a 45 min drive from Banff.
If you get a "big 3 pass" those 3 are included. Most employers in town offer a discounted big 3 pass, which is key as it's close to half-price, $700ish down from $1400ish for a midweek mon-fri pass. Also a lot of them offer corporate passes which you have to book through the employer. Caribou properties is a good place to work, I have lots of friends who have been there for years.
As for getting to the hills, a bus pass is a HUGE rip-off (unless you work at a resort then it's like $20/mo). Daily round-trips to Sunshine are $15, again, a huge rip-off. How comfortable are you with hitch-hiking because that is often your best bet. I pick people up all the time on my way to work at Sunshine, and once on my way to Lake Louise. People around here are pretty laid-back so it's always nice to have some company. Best option however is to make friends with people who have cars and buy them coffee / beer for rides