What is the best way to get from Eaton Centre to Casa Loma? We won't have a car.
Walk a couple blocks west to University Avenue and take the subway from Osgoode Station to Dupont. Get off at Dupont and walk north. You will see the towers dead ahead when you walk out of the station.
or .........get on subway at Eaton Ctr either Dundas or Queen and go south .... it will down and up the loop and as Richard said .... get off at Dupont. Just in case you want to save a 10-20 minutes. Not sure of age or ability so rather be safe than sorry.
Others have already given you a good idea of how to make the most efficient use of of the TTC to reach Casa Loma from the Eaton Centre. On the off chance that you might be open to a more scenic though slightly more complicated route, you could also use public transit - a bus and/or streetcars - for a slightly more complicted though much more interesting trip through TO's Chinatown/Queen Street West and Annex neighbourhoods:
Route 1 through Chinatown and the Annex:
catch the bus westbound from the corner of Yonge and Dundas (the north end of the Eaton Centre); get a transfer, and change onto the northbound streetcar at Spadina for a quick trip through the end of Chinatown and the south Annes; walk north along Spadina from Spadina Station to Dupont Street and station - you'll see the castle towers at the top of the hill (actually a "scarp").
Route 2 along Queen West, Chinatown, and the Annex
basically the same as above, but substitute the Queen Street streetcar from Queen and Yonge in place of the for the Dundas bus; everything else remains the same including cost and travel time
I'm not usually on the clock when visiting Toronto, but I'd say either route would take about an hour's travel time. Certainly it's longer and more complicated than riding the subway, certainly, but if the journey is just as - or more - important than the destination, either of these routes will give you a street-level view of Toronto.