When you say "do" Banff, I presume that means you're going to stay in the town and visit the local resorts (Sunshine, Norquay and Lake Louise all fall under "Banff")?
Norquay is very close to the town, about 10-15 mins drive but if you're not a beginner then you'll get bored in a couple of hours, it's very small and other than the runs from the North American chair (ungroomed blacks - both times I went last year the top was mix of gravel and snow) the runs are very short and easy, but there are a couple of steeper sections. Queues on weekends and holidays for the North American will be long and slow*
Lake Louise is an hour away and probably the closest to the size of a standard european resort, with plenty of lifts and some stunning on and off piste (and scenery). My favourite resort I went to last year. The Back Bowl is nice and easy to ski and no need to hike for most of it, and even a few days after a dump you can usually find lines with fresh powder (especially straight off the back - mix of single and double diamonds). The Rock Garden is fun, but only do it if there's plenty of snow and you're not too worried about scratching your skis/board.
Sunshine I haven't been to yet, but from what I've heard it's close to Banff, but once you factor in the gondola up to the main ski area it's about the same time as Lake Louise to "get" to. Also not great for snowboarders as theres a lot of flat sections apparently.
There's also Nakiska which is about half an hour or so from Banff back towards Calgary (in Kananaskis), which is supposed to be a nice hill, less off piste and harder/icier runs - it's regularly used for racing competitions.
Other than resorts there's the obvious, Lake Louise Lake (various long and short hikes around to viewpoints, next to the hotel, Banff hot springs etc.
Food wise Sleeping Buffalo Restaurant is really nice, as is apparently the Grizzly House.
*The lift was modified last year for pedestrians to get up to the restaurant at the top, which means it has a lot less capacity than normal lifts. It's also the only lift that services the more challenging terrain.