I need to paint over woodwork and don't know if it is oil or latex. What should I do?

If you are using latex, which I recommend, it won't matter what the old paint is. Make sure the old woodwork is clean, scape and scuff with sand paper where needed. Paint with a good acrylic latex paint.

Does the paint peel off or flake off, if it peels its laytex if it flakes off or if you sand it and it creates dust its oil.
You can also prime it and use what ever you want to paint with then

Can I use latex primer over oil paint or do I need to use oil primer?

Yes, once the surface is clean and run over with a tack cloth go with a good latex primer. Do not use an oil primer, if the old paint was latex to start with oil will not adhere properly.