I like Gunzilla or MPro7 for cleaning because they don't have any nasty fumes like most petroleum-based solvents, but just about anything will do the job.
You can also make your own "Ed's Red" by mixing equal parts ATF, mineral spirits, kerosene, and acetone. This is at least as good as any of the commercial petroleum-based cleaning solvents and is far less expensive.
Grease is best for lubricating sliding surfaces; plain old automotive grease is just fine. For oil, just buy a quart of ATF or synthetic motor oil and you will have a lifetime supply. Cleaning kits often come with a small bottle of oil. Keep the bottle after you use up the oil that came in it and you can refill it with the motor oil or ATF.