There isn't really much right down town. As Michael has pointed out, you need to belong to a club or you can use the pool at Scuba2000.
If you are a student at the University of Toronto you are a member of Hart House. For an extra $52.80 a year you can join the
Hart House Underwater Club. Membership is from May to April. If you are not a student at U of T, you'll have to get a Hart House membership ($170/year) plus the HHUC membership.
Most pool facilities don't allow scuba diving training for various reasons: (a) students dropping weights damage the pool, (b) dirt gets dragged in with scuba equipment, (c) don't know anything about scuba and fear the legal implications. The few pools that will allow scuba training do so with the understanding that the business conducting the training (dive shop) will be accountable for damage to the pool, clean up costs and legal implications.
So, basically, you'd have to find a club or dive shop that uses a pool and join them. Sadly, the only dive shop in the downtown area was Tam Dive. They went out of business around 2004 and HHUC bought up all their equipment.
If you do find a place downtown, please post it here. I've been looking on and off for the past 3 years.
P.S. Aquarius Scuba on Dundas (west of Keele) is probably closest store, Scuba2000 (highway 7 & 404) would be next and AquaSub (Bayview north of highway 7) would be third closest. I prefer Happy Divers Den (Mississauga), AquaSub and Aquarius, in that order.