Yes, public transit for Toronto is entirely reasonable - but not from Oakville, not unless you're right near the Go station (and even then, those trains don't run that often).
(I'm good for Toronto and Montreal, not much use for the rest of the country.)
Right, areas. What are you looking for? The area I like best is right near Dufferin, but accommodation there is, well, rare. We have a weird rent-a-family's-house deal, which works very well for us. There's a playground called Dufferin Grove which is my favourite playground in the world (I'm sure I've mentioned it on here before!).
The Beaches is good. It's nice out there.
Are you thinking self-catering? I'd look on vrbo or similar and see what you can rustle up and then consider areas. The bits near Bathurst and Bloor are pretty good. Oh, look, here is a neighbourhood map for downtown. I wouldn't stay in Downtown West or East, or St Lawrence. I'm not sure about the eastern bits here - Cabbagetown is meant to be good, dunno about the rest, they might be a bit grotty. All the bits to the west are nice.