M.M.
Have you tried sippy cups with soft spouts? My son never took the hard spout ones or even medium soft ones but likes the soft spout sippy cups(we use Little Munchkin). Now he can drink just fine from hard spout sippy cups but not interested in taking more than a few sips. I give him his soft spout one and he finishes his milk and asks for more. From my experience, you will have to try a few sippy cups until you find the one he takes.
I would get rid of the bottles now. I don't know how different it is drinking from bottle vs sippy but I would suggest stopping bottles right now before it becomes a habit and he won't let you take away his bottles. It's easier to do it now.
At my son's daycare , they started introducing regular cups at 14 months or so and when he moved to a toddler room at 16 months , they did not have sippy cups anymore. He still uses sippy at home.
If you don't want to transition to sippy and then to regular cups, now may be a good time to get him to start learning to use regular cups. Yes, he will spill a lot and lot of mess, but will eventually get it.