99% of the kids playing at chess tournaments have this set http://www.uscfsales.com/chess-sets...-pieces-vinyl-chessboard-deluxe-bag.html
You can get single-weighted, if you are afraid that this one is too heavy (and it probably is for a 6yo chess player).
You definitely don't need something more expensive now, but later, if you'll decide to go to a tournament, you'll have to invest in a good chess clock.