Problem solved guys. Got the alternator tested at a local alternator rebuilder and it turns out it wasn't charging at all, even with the new regulator. Luckily they already had a rebuilt alternator for my car in stock. Got home, installed it, charged the battery for an hour or so and the engine roared to life. Started fine the rest of the day too. Battery light is out, the tach works fine now, so no more problems.
I asked the guy at the alternator shop why it wasn't charging even with a new regulator. He said that the windings, separated by thin insulators, in the stator probably melted and shorted together from when I mixed up the battery connections. Showed me an example of it.
Moral of the story, pay attention when you're hooking up a battery :lol: .