In Bangkok you have to see the Grand Palace, Khao San Road and some of the shopping malls like Siam Paragon. If you are really into shopping visit Chatuchak Market as well.
On Phuket be careful which beach you choose, Patong will be very crowded and commercial, the other beaches will be more quiet. Up to your own preference really.
Why not visit Koh Samui? I personally like it better then Phuket.