I live in Downtown Toronto on Bloor Street West, and I love it here. I'm about 30 seconds from the Dufferin subway station and I can get around wonderfully without a car.
For apartment listings you should also try the globe and mail:
http://www.globeandmail.com I'm not sure if they have their classifieds online but I think you can pick up the paper anywhere in Canada.
Another way to find a place is to just walk around parts of town and look for "apartment for rent" signs. I always see a few up in the store windows around my house. This might not be practical if you can't get down here to check things out.
My girlfriend and I looked at about 30 apartments this summer before we found our current place - it's an apartment over a butcher shop, which sounds bad, but the place is amazing. They just built it, so everything's brand new. Rent is $1200 per month, it's got central air, natural gas heating, a huge kitchen, two skylights, and our own (non coin) washer and dryer. I looked at some places for $1400 that didn't have half the things this one does, so it really pays to search around.
I'd recommend living somewhere on the subway line as opposed to the bus line. You can always tell in ads because if they say "Close to TTC" rather than "Close to Subway". The major bus routes are OK, but two years ago I lived on Avenue Road, which is a pretty major road, and the busses only came 3 times an hour. It just meant that if I wasn't outside at 8:17 8:37 or 8:57 I'd be waiting for the next one. I'd say it's always better to live a little farther away from downtown if it's close to the subway.
Speaking of the subways, the metropass (bus/subway/streetcar) costs $98.75 per month, so it's kind of pricey. Still beats owning a car I guess, since I don't like driving.
As far as good parts of town, I don't really know the bad/good neighbourhoods. As KerryK suggested, the Star has a really great online classified search engine where you can specify how many bedrooms you want, what else you're looking for, and what part of town.
Feel free to send me a message if you have any other Toronto questions, I'll be happy to help you in any way I can.
Bob