I think you're confusing a few things here. Antimatter has nothing to do with anti-gravity. Antimatter exerts the same gravity as regular matter.
You're talking about exotic matter, which (theoretically) has negative mass and therefore negative gravity, but this has never been observed. Related is the concept of exotic energy
If we can learn how to create and manipulate exotic energy it's a doorway not just to anti-gravity but to the possibility of creating wormholes or other FTL travel techniques.
And airplanes don't count as anti-gravity since they don't negate the effect of gravity outright, they just exert equal for in the opposite direction. Anti-gravity is often related to the sci-fi idea of a "reactionless drive", a method of propulsion that requires no propellants because it manipulates gravity to create motion. Such a drive would appear to violate Newton's first law but I believe general relativity doesn't completely preclude the possibility of such a drive existing.