Spanish River
The Spanish river is a highly recommended experience. Overall, it's the best river for introducing longer whitewater canoe tripping. The Spanish has plenty of class 1 and 2 rapids that go on and on. Many of the Spanish's long rapids are easy enough to run with young kids aboard. The draw of this river is the rapids, the views, and the cool experience of the train. A lot of the campsites are a little lacklustre though and many of them are right in the middle of a portage.
The Spanish River has two main branches that are both named Spanish River. One of them, the Duke Lake access point, is accessible by car, and the other branch is only accessible by train—which is pretty cool.
VIA's Sudbury to White River line (known as Budd train) still operates and offers one of the best canoe shuttle experiences in Ontario. The train will meet you in a town called Cartier where you can load up your canoes and gear and get dropped off at your choice of put-ins, all depending on how many days you want on the river. You'll need to use the old-school telephone to pay for the canoes (VIA: 1-888-842-7245) don't bother with the online booking for that.
Split Rock added the campsites to Google Maps to make planning easier for everyone.