The best grease I know of that won't migrate and won't dry out is 'vacuum grease': it's made for greasing laboratory glassware fittings that are subject to vacuum and high temperatures. It's not much on extreme pressure - extreme pressure as in heavily loaded bearings. The stuff is expensive, though and you have to buy a big tube. A good alternative is moly grease made for brake calipers - you can buy small (good for one brake job) tubes at the auto parts store for a buck - this is a pretty thick grease good for large gears and winding mechanisms.
As to oils. Well, you can do a lot worse than plain-old SAE 30 oil as you put in the car.
For oiling delicate mechanisms - think clockwork in shutters and self-timers - the best is Nye Clock Oil. However, 'Turbine Oil' from the hardware store is also very good stuff.
Be very, very sparing with lubricant. Apply oil with a toothpick, only the thinnest film is needed. Ditto grease, wipe some on, wipe most of it off.
Naptha - aka Ronsinol or lighter fluid - is the best stuff for removing the old lubricant. It dissolves caked-on grease, won't harm plastic and it evaporates without a trace.
Do not ever use WD-40, LPS-25, 3-in-1, Liquid Wrench or any other product that comes in a spray can or advertises itself as 'penetrating'. These products are great for freeing rusted up 1/4 inch bolts and garden equipment. They all, without exception, turn to a gooey gummy mess in short order.
If in doubt, or if it feels like you are forcing something, then stop the DIY and send it off to Mr. Wong. The camera can be close to 40 years old. After a good strip and clean it will be good for another 40 years. Like a Leica M3, an F2 can be kept going forever.