My kids were both around 20 mos. It seemed for them they'd reached a point where, if they couldn't see it, they didn't want it so it was a natural transition for them. I just never let them see a bottle again.
They'd cry a little the first day, but by the 2nd and 3rd day, they were over it.
One word of advice-sippy cups are very popular, but there is some evidence that they can hamper speech development. Straw cups are best. They cause the child's lips to really pucker and round, which is really good for the mouth muscles.
My son has a disability and no one had told me that. He already had a speech problem, and it was made worse by using a sippy cup. Most kids don't have a problem and sippy cups don't affect them like they did my son. But I do tell people this, just in case. Good luck!
Also, when we did away with the bottle, I fed my kids more food, so they weren't getting full by drinking a bottle of milk. That seemed to help too.
ETA: I don't think there is evidence that sippy cups hurt speech development. Our SLP told us that if a child has weak lip muscles, they won't help strengthen them like a straw would. It's much harder to drink from a straw than a sippy cup-you use more muscles. My son couldn't drink from a straw until he was 4, which was when he finally was able to pucker his lips.
I think what I wrote above about hamper speech development is incorrect and misleading. I apologize for that.