There's plenty to choose from. Couldn't tell you what the "best" is. I use Mobil 28, I think it's called.
You'll need a solvent to melt away the old grease. Degreaser ought to suffice, does for me, squirted from pressurised spray cans in a narrow jet it gets rid of the grease easily enough, with an old toothbrush to get rid of the big blobs.
Once the gears are clean, inspect for mangled and/or cracked teeth. Cracks can be small, magnifying glass might help.