Determine the type of wood your swing set is made of. Some, like redwood, will likely require a stain or varnish, while others can be painted over.
Mark any portions of the wood that you need to refinish. This includes areas where the paint has peeled or the finish has come off, places that are rotting and places that show signs of splintering. This is where you need to focus a little more effort than the rest of the swing set.
Use sandpaper on the areas where splinters are visible. Apply the sandpaper until the surface is no longer splintering and is smooth to the touch.
Apply paint or wood stain to the areas where it has peeled or come away. Specifics will vary depending on how much damage your swing set has incurred. If you are using paint, ensure that the new coat is even and blends well with the existing paint. If you use wood stain, the color should be even with the rest of the wood.
Allow the paint or stain to dry. If you are using wood stain, you may wish to sand the area with sandpaper afterward and then apply a coat of varnish to help protect it. You can also repeat these steps multiple times, applying multiple coats of stain and varnish for better protection.
Check the metal bolts and screws on the swing set. They may be showing signs of rust, in which case you should replace them with new ones. This is especially true for the swings themselves: look at the chains, the seats and the "S" hooks to make sure they haven't rusted or corroded.
Unscrew any rusty screws and replace them with fresh ones. This may require you to partially disassemble the swing set.
Use pliers to remove any rusty swing chains or swing seats and replace them with new ones. Most swing chains are connected using "S" hooks, which you can pry open and seal shut with pliers.