The first thing I would build would be a perpetual motion machine. With gravity operating on one side of a vertical wheel and "anti-gravity" operating on the other side, the wheel should turn by itself. If the energy required to power the "anti-gravity" sub-system is too great, then perhaps our expectations that this "science" can eliminate rocket engines will have to be re-examined.
The existance of anti gravity wouldn't imply overturning conservation of mass-energy. Remember too that folks said that rockets would never be able to take people to the moon, due to "practical" considerations.