Join Date: Mar 2005
Location: Calgary, Alberta, Andorra.
Interest: hiking, backpacking, scrambling, climbing
For hiking/backpacking, Don't Waste Your Time in the Canadian Rockies is an excellent guide for suggesting the trips to make best use of limited time.
But early June is still barely spring here, so there's a lot that can't be accessed yet.
Definitely the icefields parkway will be not yet hikeable.
pick up the Banff GemTrek map, and look for south-facing trails.