how to cut off bottle?

ftm. my daughter just turned 1 a few weeks ago. The past week, I've been trying to cut her bottles off during the day. i give her a bottle before bed and in the morning. She doesn't sleep through the night yet and wakes up for bottles (at least twice). During the day, I've been trying to put her milk in her sippy cup.. But she only takes a few drinks. She's a thumb sucker and when I notice her sucking her thumb alot, I give in and give her her milk in a bottle. I feel like I'm going about this completely wrong. I don't want to "starve" her.. how do I do it?