I'd definitely recommend exploring Queen West. It's an interesting walk all the way from the core west to Ossington and beyond. Lots of cool shops, restaurants and bars. Ossington's worth checking out too.
I'd also consider taking the ferry over to the Toronto Islands. Nice place for a picnic.
The campus of the University of Toronto are quite large and picturesque. Lots of very old and beautiful architecture to explore.
For shopping there's the Eaton Centre, as well as the Yorkville neighbourhood and Bloor Street between Yonge Street and Avenue Road.
Some of the best pizza places in town are Terroni's (there's one on Queen West) and Pizzeria Libretto (on Ossington) - try their margarita pizza. As far as chains go, I like Mamma's Pizza. Also try the Mad Italian on College Street for their pizza cone. Speaking of College Street, it's got a lot of good Italian restaurants, like Teatro.
For a good breakfast place, try
Cora. It's a Canadian chain with a couple locations in Toronto.
If you're interested in Thai food, I recommend Khao San Road on Adelaide Street, but be warned that there will likely be a lineup to get in.