Gravity is a force. If you don’t believe me, grab a brick, hold it over your big toe (not too high, I don’t want you to injure yourself) and release it. After you’re done yelling in pain, tell me how it is not a force.
To refine this statement: In general relativity, gravity is considered, much like the centrifugal force, a pseudoforce, which arises because when, say, you are standing here on the surface of the Earth, you are, in fact, in an accelerating reference frame.
Now gravity is universal: it acts on all forms of matter equally. Therefore, if I were to apply a geometric transformation to cancel out gravity, it would cancel out gravity for everything. This allows me to interpret gravity as geometry, especially so since the quantity representing the gravitational field does play the same role as the so-called metric. So yes, gravity is arguably geometry.
But it is also a field that carries energy and momentum, and it can be nonzero far from any sources, where its energy and momentum propagates in the form of waves at the vacuum speed of light. So yes, gravity can be a wave.