If you have your bikes with you, definitely explore the downtown core. You can find some really nice little neighbourhoods.
And yes, Toronto has some great green spaces.
Check out the "gaybourhood", Church and Wellesly streets. Check out Parkdale, it's kind of sketchy but has some interesting history and some pretty cool bars.... not to mention big old houses and the lakeshore trails are just south of there.
I'm not familiar with cheap places to stay because I live here, but I'm sure you can find something. It's a good time of year to be around that's for sure.
While you guys are here you might want to consider renting a car and going up to Georgian Bay (Killbear park comes to mind, I go there every year) or Algonquin. No sense coming to Ontario and not seeing some of the great outdoor places we have.
Enjoy your stay!