I use the mobil 1 red grease right out of my grease gun. I spray the door roller axles with lithium spray grease. I think genie had some sort of lubriplate grease, but the red m1 grease works fine on mine.
From a Genie screw drive opener manual...
"Lubricate door rollers, bearings and hinges: Use silicone lubricant or lightweight oil.
Lubricate the drive screw: Wipe off old grease using a lint-free cloth. Apply a light coating of silicon based grease. For best results, use Genie® GLU-R." http://www.geniecompany.com/data/products/dealer/trilog-1500-install_new.pdf
You can't go wrong using a bicycle chain lube for the chain. Personally, my choice is simple: For uncomplicated lubrication needs such as this, I'll use the best, most appropriate lube I can find, regardless of cost or difficulty obtaining it....so long as it's whatever I happen to have in my garage that day. And that would mean grease -- whatever I've got -- for the screw, and random motor oil for the chain and door hinges. I'm a cheap b*staldo! I have a foolproof method of determining the correct lube for a task....If it's in my garage, and therefore "free," by definition it's the right lube.
I use the same NGLI#2 grease on the garage door tracks and internal sprockets that I would use on a wheel bearing. On a chain, any light lube works fine. If you have bicycle chain lube, that works great.