Carpentry and Woodworking - Installing baseboards

The baseboards in my apartment are very old and the wood is badly damaged. How do I remove the old baseboards so I can install new ones. What tools should I use? I am very worried about damaging the walls.

I would suggest that if you don't own the apartment that you should FIRST contact the landlord and get permission, preferrably in writing. Your lease may state that you cannot alter the apartment in any way. How badly damagaed are the baseboards? Could you spruce up by using a little kilz, wood filler and/or paint?

I do own the apartment so there is no landlord to help out. The wood seems to be rotting away in places, it is chipped and there are so many coats of paint that much of detail is gone. Would it be better to strip the paint?

I'd strip the paint, fill the gouges with a wood spackling, sand, prime, and repaint.

You say the wood is rotting away. Is moisture present?
wood only rots when moisture is present. I think I would remove a piece and find what's really going here.

We have had somewhat luck at scoring the paint with a sharp knife and/or a flat wide trowel where the wall meets baseboard. You may have to work at this with so many coats of paint! Then prying the baseboard off the wall. We've then had to scape/sand/fill the areas on the wall where the paint is built-up after removing baseboard to even out wall. We've also ended up having to replace some portions of drywall and have gone as far as having to replace sections of wall studs and bottom plates. We've opened up cans of worms, so to speak, where one thing leads to another in small projects as yours. Do you know why the wood is rotten in certain areas? Good luck with your project.