When buying and playing a Villa in Dominion do you in a sense get 3 buys for that turn?
You don't get a new free Buy, sorry. Nice thinking though.

The base game rules do say that when you enter a Buy phase you get one default Buy, but the card-specific notes for the Villa in the Empires rules (page 11) clarify (or revise?) that you only get one default Buy in the entire turn, not one each buy phase:


  If you buy Villa, that uses up your default Buy for the turn, however playing Villa will give you +1 Buy and so let you buy another card in your second Buy phase.


Another way to think about it might be that the game gives you +1 Buy as you start your turn, but not each Buy phase as the base rules might imply.

Once you re-enter the Buy phase you will have 1 Buy left from Villa itself, in addition to whatever other action cards you've played this turn, but not 1 free Buy from the game. From what you described you only have 2 Buys this turn.