From Vancouver, you can drive out to Calgary in about 13 hours. Even with meal and washroom stops along the way, you can get there in a day. Of course, if you’re wanting to sightsee and you haven’t gone before, then taking several days may be a better course for you.
You can also take the Greyhound Bus which does stop at various pickup/drop off points which does delay the time, resulting in 16 hours. But the stops are short, so this is NOT for a sightseeing option other than seeing the scenery through the window.
Best time of year to go is between June and August. I remember still in May, the lakes can be still frozen (like at Morraine Lake) and weather can be pretty unpredictable outside of those summer months. Depending on the season, you’re bound to see some wildlife such as Elk, Big Horn Sheep, Mountain goats, eagles, porcupine, coyotes and if you’re lucky, some bears. I have yet to see a cougar, fox or a wolf. (see my post on a guidebook on the animals and birds of the Canadian Rockies that I illustrated and my dad published)
For me, I grew up in Calgary (1 hour and 30 minutes from Banff) and my dad ran a tourism company, so I went to the Banff National Park often. I also lived in Banff over a summer when I was a student, so the Rockies are like my backyard. I’m not the nature-camper-hiker-snowboarder type, but I do enjoy the serenity and peacefulness. It’s especially nice after living in congested, loud downtown of Vancouver to go somewhere that is the complete opposite.