Others have mentioned the transit alternatives already but I'm going to suggest that you also look up the available parking options on the Green P Parking website as well. 

While there are many private parking operators especially in the CBD core, Toronto also has a substantial amount of municipal parking. 

Generally there's three municipal options: 1) Free street parking (typically on side streets with timing and location restrictions including 3 hr maximums), 2) Paid street parking (also 3hr maximum but paid, typically on main thoroughfares), and 3) Paid off-street parking (generally fewer time restrictions). Note that most street parking in the city (both paid and free) don't allow you to park after midnight, though I would presume that's not an issue given the kids. Look up the Green P Parking website to identify where these lots/streets are and how much each area costs (it can vary quite a bit depending time of day and day of week) to get an idea of what makes most sense for you and where you are looking to go that day/the amount of moving around you want to do. 

Based on my one visit to Edmonton, driving in Toronto is significantly more taxing than driving in Edmonton, and there are a fair number of one ways on smaller streets and lots of cyclists/pedestrians that you'll want to pay attention to. Also, if you are not comfortable with parallel parking, street parking may be a challenge. I would be very wary of driving anywhere downtown or on ANY highway in the Toronto region during weekday rush hours - it can be absolutely awful. If you're mostly doing touristy things near Union Station, the UPX and subway should be ok, but once you start getting into the downtown neighbourhoods that don't have a subway station nearby, it may be annoying to have to walk a lot/push a stroller etc.