Is there a specific tread depth at which it's time to replace, or just when I feel them start to slip a little bit?
Also, do I have to get similar tires on the front and rear? As in, both radials or both bias or both of the same model (just different sizes)?
There should be wear bars on your tires, they are the pieces of rubber that cross the "hole" part of the tread at various places. When the rest of the tire gets down to that level, it's time to replace the tire.
If you have lots of cracks, or other unsafe things happening its also good to replace your tire.
As far as matching your tires, everyone will tell you something different. Personally, I'm of the mind to match the front and the back. Then you know that they will behave similarally. If you do decide to replace one of your tires with something different, make sure that you don't have a stickier tire on the rear than the front. You do NOT want to make it easier for the front to slide when the rear isn't.
As far as what tires to get, use the search function. There are a LOT of threads here about tires. Avons and BT45s seem to be among the favorites.
There is another thread on the main page that talks about tire life (http://gstwins.com/forum/viewtopic.php?t=17343).
If you are trying to get the most out of your tires, its best to replace them the day before you have the accident..
I'd recommend you replace both tires as soon as one of them shows less than 2 millimeters tread tepth at the center. I never checked my original equipment Bridgestone Excedra's but when I swapped them out with BT-45's the new ones showed a 7 mm depth.
Loosing traction in a wet corner is no time to realize there isn't enough tread left on your tire. ;) Better to replace them with some bite than none...