Hi, Donna. I believe that your first order of business is to find out what kind of wood the mantle and surround are made of, if you have an older house. If you have oak or something under the paint, you might be better off stripping and refinishing since the oak will kind of hide the future dirt.
If you have pine or similar, you might just want to repaint. No need to strip unless you have a heavy paint build-up. One part of the secret to future success is to paint in a color that will mask future dirt. If you paint with a gloss paint, this will make future cleaning easier but will also show dirt sooner. You might consider some kind of faux paint finish that, again, will hide some dirt.
Once you have direction on how you want to proceed, preparation is a major part of getting a good paint finish. The second part is buying good paint from an independent paint store like Sherwin-Williams or Benj Moore. If you are going to paint again, wash with something like Dirtex. Then use a primer that will hide and seal any remaining dirt, like B-I-N. The primers that most paint stores sell is nothing more than thinned paint. Primer/sealers are much better. When you have the priming done, you can proceed with the paint that the local independent paint store suggests.