In recent years there have been a good amount of surveys and publications on "computational" or "combinatorial" geometry, and looking at them may give you a good idea of current questions. Specifically, there is the excellent recent book "Research Problems in Discrete Geometry" by Brass, Moser, and Pach. You may want to start by looking there and at the references it provides. Besides a good deal of information on classical questions, among many other topics, you find:
- Density problems for packings and coverings.
- Distance problems.
- Lattice point problems.
- Graph drawings and geometric graphs.
- Geometric inequalities.