Hi J.,
Both my kids went cold turkey from bottle to cup. My daughter (older) had a bit of a battle of wills, but we basically did what other responders did and hid the bottle, offered only the cup, and emphasized "big girl". Took about a day and she was all cups -- however, I soon noticed she was having some trouble, like she was always trying to suck off the spout of the cup. So, I tried a cup with a straw, and she was happy with that. My son went straight to the cup, no problem, but I think he modeled after his big sister (though he uses a sippy cup, not a straw).
One thing with my son -- he refused milk in the cup, too. He'd drink water fine, but would absolutely turn away from the milk. At first I thought it was the cup, but it turns out that it was milk -- just didn't like it. So I asked our doctor and she recommended trying soy milk because it was sweeter and some kids just don't like the taste of milk. So I tried that and he loved it, so we used that for a while (I'd actually mix a little cow milk in with it). After about five months, I saw him steal his sister's cup and drink regular milk, so I said, okay, try your own cup! and he's had regular milk ever since. Don't know why he suddenly switched, but he did! Hope this helps & good luck!