I started letting Julia drink from a sippy cup around 6 or 7 months. She was eating a lot of solids pretty early and was getting constipated, and I thought extra water might help. It didn't, but that's another story. She liked the cup and lot and regarded it all as a big game, so I let her keep trying a little every day.
Anyway, I could not figure out how to teach her to suck from the spillproof ones-- it's a much different kind of sucking than sucking on a nipple-- so I either used the Avent trainer cup and left out the spillproof thingie, or I used the First Years cups and broke the spillproof valve with a knife so that the liquid flowed freely. It is messier that way, though-- some of the liquid inevitably goes down her chin. She still doesn't drink out of it independently though even at almost 10 months; I need to hold the cup for her if any real drinking is going to happen.
I also taught her how to sip from a regular plastic or paper cup, and how to drink from a straw-- I put the straw in a cup of water and put my finger on top to trap the liquid, and then put it to her mouth, and she sucks the water out of the straw.
I have heard of babies as young as two or three months learning to sip from a cup, when they were refusing bottles. I think you just have to keep trying a few minutes a day and they get the hang of it eventually.