It all comes down to the age and the materials used. A picture would be fantastic to provide a bit more help
If it’s the old white oak chair and desk, I’d use a product that melts the shellac back from the cracklings phase. If it’s the metal type with plywood top, I’d sand the metal, prime and paint with rustoleum.
Not sure exactly what style you have, but I thought this was a fun makeover that could be inspiring: https://www.meatloafandmelodrama.com/vintage-school-desk-makeover/
You can just paint.
We love this formula an salvage shop gave us for wooden doors. It has since worked on a variety of wood projects:
1/3 boiled linseed oil
Does double duty cleaning and restoring/finishing.