The Toronto Zoo is located quite far from downtown, and is not possible to get there by subway
The best way to get there is to drive. Taxiing or Ubering there will be quite expensive
The best way to get there by public transportation from downtown is to take the GO Train Lakeshore East line.
Get on the train at Union Station (or any other station along this line) going east to get to Rouge Hill station (if you're leaving from downtown's Union Station, follow the signage TVs to go to the platform for Lakeshore East. If you're leaving from other stations along the GO train line, just make sure you get the direction right for Rouge Hill station).
Once you get off at Rouge Hill station, get on a TTC (the red buses you'll see on the surface level that's part of the Toronto Transit system) route 85 Sheppard East (take the west-bound bus of this route), and this bus should take you all the way to the zoo. Bring cash or buy some tokens for the TTC bus. They won't take credit card on the bus.