I have the RC54, but it's basically the same, mechanically.
I typically grease the spindles every 2-4 hours of use. I pump until I hear the crackle of escaping air and grease coming from the upper seal, which means any more pumping and I'm just pushing grease out of the cavity. When I did it yesterday, the right spindle took 4 pumps and the left took 5. Don't forget the belt tensioner pulley! These deck spindle castings hold a lot of grease, so there is some reserve capacity. But, the spindles tend to run fairly warm and at high speed, so I like to keep them well-greased. [When I bought this tractor, maintenance had been neglected and the bearings in the outer spindles were absolutely SHOT!]
As for the gearbox, it's a good idea to check the oil level frequently. These have a reputation for wearing out the lower seal, and if that happens, all the oil will leak out and you're likely to ruin the gearbox. Right after use, when you park the machine, place a piece of cardboard on the floor beneath the center spindle. If you see any spotting from leaking gear oil, that's a signal to be extra vigilant. Not a bad idea either to just lay your hand on the gearbox after use to detect excessive heat. In summary, let the gear box do the talking!
At the end of each mowing season, I take the deck off the tractor, flip it over and give it a good cleaning. Then, I coat the bottom side of the deck with used motor oil to store it over winter. That's a good time also to check the gearbox oil level. And, to grease the driveshaft joints. And, to sharpen the blades.
Yesterday, wifey started mowing the yard, but forgot to gather up the dog chain beforehand. It wrapped up around the center spindle so tightly that I had to remove the blade to get it cleared. She said smoke was pouring out from under the hood. I found that the two drive belts at the front of the engine got flat-spotted. And, then I learned how to adjust the belt tensioner pulley at the front of the engine. Put the belts on the list for next time I'm down near Messick's!
-Paul