Babies can start drinking from a sippy cup from as early as 6 months of age.
By 9 months you should start replacing the bottle completely with a sippy cup and then once they get the hang up the sippy cup usually by 1 year of age or after then they will be ready for a regular cup.
You should break your baby from the baby bottle as soon as possible because if you don't you'll have a harder time getting the baby to give up their bottle for the sippy cup.