If you are moving to Mississauga then no one can help you. You can help yourself by doing a search in the 411 directory or pick the yellow pages and turn the pages.
Cheapest way - rent a truck from U-haul, then put an ad on Craigslist/Kijjijj for a couple of helpers for $10 - $12 per hour. Just have to manage them well, and be prepared to chase away anyone not suitable when they pitch up
I do not know if this exist in the GTA, in Montreal they have a service where they cart wood storage box to your place and put it in the driveway or the lawn. These boxes can be up to 10 x 10 x 10 ft. There are smaller ones.
You load it, you unload it at the next destination. They can also store it for you. Just have all your stuff boxed up and have some. local kids load and unload for you. If you want to be enviormentally friendly and not use cardboard boxes, they also rent plastic boxes for you to use in your move.
I used 2 Men and a Truck (Midtown) this year. It worked out fine.
There ought to be another franchise in the west end somewhere. You'd be better off with the nearest operator because they bill from the time they leave their yard until they get back there, and not just for the time spent moving between two points.