If the swing has already been stained before you might want to use a gel stain. Just clean it well and give it a light sanding first.
Buy the wood stain of your choice at a home improvement store. Brush it on, but then use a rag to finish it to keep the brush strokes off. This is the easiest way to get the stain even.
It will need to be cleaned first if it has been outside. Before I stain anything except new wood, I go over it with fine steel wool to prepare the surface. Clean with denatured alcohol to remove any residue. Make sure your stain is thoroughly mixed. You can apply with a brush or a rag, I prefer the rag method. Wearing gloves, you apply the stain and then take a rag and rub it off. The longer you leave it on, the more it will seep into the wood and give a darker color so you might want to test time on the bottom side of the swing. Work in long even strokes to keep the finish as uniform as possible. Once finished staining, you will want to seal it since it will be exposed to the elements.
Another option is to purchase a stain that has a finish mixed in it. In that case you will brush on and leave. I did that for my laundry room cabinets.