From my reading Wikipedia, I understand there are several branches of classical geometry (if the ordering is off, or I'm missing a few things, let me know):
- Absolute
- Euclidean
- Non-Euclidean
- Spherical
- Hyperbolic
- Projective
- Affine
- Transformational
- Inversive
These can be studied from:
- Coordinate geometry (aka algebraic) view
- Axiomatic (aka synthetic) view
- Analytic view
The combination of these branches and points of view generate many possible avenues of study.
How many of these are useful for a modern geometer to know? By useful, I mean fairly likely to lend intuitive insights into their work (This may seem vague without specifying what kind of work they do. But since most geometers aren't working in classical geometry, I guess the work of any such geometer could be considered relevant to the question).
How deeply must they learn?
I've got a good high school level knowledge of math, and some college level math, and I have an interest in pure mathematics for a career.
EDIT: Added tags relating to modern branches of geometry, so that these kinds of geometers see this question.
If you are planning to do research in modern geometry, pretty much nothing on your list is relevant. Instead:
If you are planning to work in algebraic geometry, you should learn commutative algebra or/and complex differential geometry.
If you are planning to work in Riemannian geometry, most likely you should spend time learning differential equations, functional analysis, Sobolev spaces, etc.
I know, this sounds a bit sad, but the reality is that you have only that much time. I do not mean that learning "classical" geometry is waste of time, not at all, but you have to prioritize...
Edit. Of course, if you are planning to work in hyperbolic geometry (which is an active area of research if one studies hyperbolic groups or hyperbolic manifolds or discrete isometry groups of hyperbolic spaces), then you should learn some hyperbolic geometry. But this is a bit of a tautology.