We made the transition to sippy cups when we began to give dd cows milk instead of formula. She had been having juice/water in a sippy since 6 months, but didn't like the idea of her milk being in there. At 11 months, we started with the first bottle (around 9am) and put cows milk in a sippy that was the same make but a different colour to her juice sippy. (The hv said it was ok as she still had her 2 bottles of formula, and we didn't replace it completely with the cows milk.) The first day she had nothing, but we just gave her juice and a small snack afterwards - we didn't top her up with formula as we thought that if she knew there was no alternative, she'd get used to the idea sooner. (However, we did increase the other two bottles later in the day so that she had what she needed). The next day, she had a couple of sips, the day after a bit more etc. It took a while, but when she was drinking the full cup, we started on the next bottle and did the same. When she was drinking that, we swapped the night time bottle (the one I was dreading!). She was a little over a year when we did that.
Hope this helps. I've read it through and it sounds very confusing, but feel free to ask about anything that isn't clear.