When do you plan to visit?
Doing tours to see places is going to wind up costing you quite a lot, for four people (even if three are children) but it can be done. Check the page for the Banff "Roam" transit for information on getting around in the immediate Banff townsite area. Tour companies are listed on the official tourism page www.banfflakelouise.com .
This past summer, there was a new "hop on, hop off" service offering transportation in a loop from Banff townsite to Johnston Canyon to Lake Louise and back to Banff. Parks Canada is also running some shuttle services - details will be on the tourism website, or the BNP website www.pc.gc.ca/banff
Assuming you are coming in summer, to get to Jasper, there is a Brewster day tour that runs something like 10 hours, with a long stop at the Icefields Visitor Centre for the sno-coach tour on the glacier and skywalk. In Jasper, the local tour companies are listed at www.jasper.travel .
There is no local transit system in Jasper, but again, assuming you are here in summer, there are beaches that are close enough to town that you could walk one way, then take a taxi (or cab it both ways). Also, Maligne Canyon and Pyramid Island are both close enough to town that you could take a cab to them. Brewster runs a shuttle to Maligne Lake and there is also one to the Jasper Skytram.