First of all, determine what type of wood is used to make your swing set. Once you know the type of wood, it is only then you can do the refinishing job to perfection. Woods like redwood are to be varnished to revive its look, whilst other types of woods can simply be painted to achieve the task.
Spot out the damaged areas of wood. The portions where paint has been peeled off or any rooting spots or any areas which show splintering will require your attention the most.
Take fine graded sandpaper and start grinding the spots where splinters have come out. Sand the wood until it is fully smooth and no signs of splintering remain.
For the areas where paint has been peeled off, apply a fresh coat of paint or wood stain onto them. Choose the shade of paint matching the existing one and make sure you paint it evenly. For using the wood stain, you have to make sure its colour completely matches with the rest of the wood or otherwise it will look very odd.
Give the paint or wood stain appropriate time to fully dry. This may require several hours to do so.
If you have used wood stain, grind the area once again when it is dried and then apply a coat of varnish to bring out shine in the wood. Applying multiple coats of stain and varnish over and over again will protect the wood for a longer period of time.
Examine the metal bolts and screws on the swing set and make sure they are fully tightened. If you observe any signs of rust, you better replace them with the new ones immediately. Same is to be done with the seats and the S-shaped hooks attached to them.
Examine the swing chains to see they have been affected by rust or not. Use pliers to detach them and then replace them with the new ones.