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):
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.