You won't be able to gain much height unless you can do a bunnyhop properly to begin with. If you are moving with any decent speed and time your hop wrong or don't get high enough, you'll slam your rear tire rather hard on the curb you were trying to clear which might pinch flat the tire or bend the rim.
Osco's suggestion is a good one. Keep your flats and lift the front tire over the curb, and roll slowly in a low (easy) gear to the curb then pedal over it. Piece of cake.
Keep practicing bunnyhops. When you can do it right, sure, clear the curb.