What kind of sandpaper do I need to refinish an old metal glider?

I would use a medium grit sandpaper.

Here is a great tutorial on a metal 1950's glider/bench:

http://www.tattooedmartha.com/2017/05/25/rusty-vintage-glider-bench-makeover/

Julie....Can you send a picture so we can get a better idea, please?

steel wool if you are painting yorself

Emery cloth is a good choice. It’s main purpose is for metal. Make sure to use a grit in the range of 150-220 or finer. Because emery cloth is made of a cloth backing you can ‘wet sand’ your glider. This just means to wet the cloth and sand. The water will act to remove sanding dust and materials which gives the cloth a longer life because there’s no dust to clog the cloth. You can’t effectively wet sand with sandpaper because the backing is made of paper. If you have a choice of grit materials go with the Aluminum Oxide because it lasts longer than Silicone Carbide.

I would use fine steel wool unless you have rust issues. Then use a wire brush for the rust only.