I don't oil or grease firing pins. At most I'll rub them down with whatever oil is on my fingers when I reinstall them.
The only two places I always use grease are the bolt lugs on a bolt action and the hinge on a breaktop gun. I like the Shooters Choice grease for these, but have of course used other things.
Speaking of other things to use...
I use all sorts of oil and grease on guns - I mean, I bought the stuff to try it out so what am I going to do with all of it? (The old formula Tetra Oil stinks like roadkill and gets used on gate hinges and anything that doesn't come in the house. If it says NEW on the label it's okay though.)
A few examples: TW25B on my Kel-Tec P-32, RIG +P stainless grease or Tetra Grease(and BreakFree LP) on my Kimber Stainless Gold Match, Tetra or Shooters Choice grease (the red stuff in a syringe) on my BHP and CZ 75B. Wilson Combat grease and oil is good, too. Militec is good. FP-10 is good. I have boxes of stuff I've collected over the years. Bacon grease would probably work if the salt didn't eat the metal.
I really don't think it matters, but some things just seem to work better for certain uses. Straight BreakFree CLP will keep them all running. So will Lubriplate/lithium grease or a synthetic from the auto parts store.