just in case you were not aware, there is no off roading of any sort withing the National Park Boundries. I think this likely applies for fishing as well, but perhaps with a paid guide of some sort.
however, just outside of the park, closest being Canmore and Kanaskis country, I am unaware of any trails but I am sure there are some.
Next closest is towards Cochrane on the 1A highway. A nice drive on that road, out of Canmore towards Cochrane. Before you get to Cochrane there is a Forestry Trunk Road (43 I think). up this road is many off road area's that are great.
next Half way between Banff and Jasper is the David Thompson Highway. 45 minutes towards Rocky Mountain House is Nordegg. Here there is the northern part of the Forestry Trunk Road. Go south from the David Thompson Highyway to the North Saskatchewan River. Great wheeling here (to the East Only)
My advice would be to take the Forestry Trunk Road from the South End and take it all the way North to near Edson. you will find more wheeling along this route than you can do in a Month. Bring your camping gear and make a 3 day adventure of it. You won't regrett it.
in and around Banff area, tourist stuff. Johnston Canyon on the Bow Valley Trail is worth the walk. Lake Louise and there is another lake up a long mountain road that is even nicer (sorry forgot name of lake).
30 minutes from Banff is the 93 highway. go up that road about 15 minutes and there is a tourist stop... Painted Pots and another canyon walk. THis is pretty cool area.
if you have never done it. Doors and top off do the Banff Jasper Highway. amazing !