If I wish to visit the towns of Northern Ontario starting from Toronto on a budget and I do not drive a car, what are my best options? Greyhound? Is there anything else? Like an organised car pool/shuttle service that runs between Toronto and some country side towns in Ontario?
Your best bet for public transport access to the Ontario North is Ontario Northland. They run scheduled buses from Toronto to the north, including Temagami (but not Hiawatha, wherever that is).
It is quite a trip, stopping at various small towns you won't get to otherwise. If you want to get to the real remote areas your best bet is the Polar Bear Express train to Moosonee - it's as remote as you can get without flying, which I assume is outside your budget.
It depends what you define as Northern Ontario and how much you want to see. Some places (Moosonee and Moose Factory for example) are train only. Many are bus only. You can fly to Sudbury on a regularly scheduled commercial flight. Heck, some people might consider Peterborough or Barrie to be northern, and you can take Go Transit (commuter trains and buses) there. You might take the train (Via or Go) to a "gateway town" and then the bus from there. I find bus service to be pretty sketchy - there might be one bus a day, and it's not particularly cheap. You are also likely to run into a bit of "can't get there from here" where you have to plan your route carefully.
Depending on your budget the service that would be the least hassle, and the most flexible would probably be bus. The train doesn't stop too many places up there unless your'e going into the interior on Ontario Northland (and that's only because you can choose where to stop), and VIA isn't the cheapest option out there.