Grease has several uses on a motorcycle. It's there to protect parts from water, to keep rubber and plastic seals from drying out, and to provide lubrication. Generally speaking, grease is just a reduced fat. In fact, you can find several companies who call their grease bases "soap."
The choice for a general purpose grease to protect from water, keep seals moist, and provide lubrication is easy. If you're going to do your own maintenance, I recommend you get Mobil-1 or Valvoline synthetic grease for general purposes, about $6 / pound anywhere. This stuff can handle temperatures higher and lower than you'll ever get near. A one pound jar should last you about 15 years.
There are two types of maintenance you must do on a shaft drive bike. At your rear wheel there are pinion gears where the spinning drive shaft motion is turned 90 degrees into the rear wheel motion. These gears are bathed in oil which much be changed after break in, and about every 10,000 miles thereafter. Honda says this gets done the first time at 12,000 miles. Don't believe it. Do it at about 1,000, and when you see what comes out you'll decide to do it again in about 250 miles to flush the rest of the garbage out.
Also, a shaft drive bike will have splines on each end of the drive shaft, and on a large ring which transmits the power to the rear wheel. These splines must be lubricated. This is a rather tricky subject, as the lubrication requirements of splines are very different from gears or bearings.
AGL 75w 90
Put the bike on the center stand. Place a large flat metal pan under the real wheel, like a cookie sheet. This process is a bit messy. Remove the 17mm bolt on the bottom of the rear wheel hub. Next, remove the inspection plug, 17mm large bolt at the rear of the hub. If your drain plug is magnetic, make sure to clean it completely. Replace the drain plug. To fill, get 70 or 80 weight gear oil, and put it into the inspection hole until it comes back out. About 160cc, about 1/6 quart. You may as well buy good synthetic oil, a quart is good for 5 changes.
You need to grease your drive splines each time you change your rear tire. This is true for all shaft drive bikes, regardless of brand. This is a real issue: ask any BMW rider, or any old-time ST1100 or GL high mileage type, and you'll likely get a whole lecture about spline maintenance. The lecture may or may not be informed, but it will demonstrate real concern.
The drive splines get surprisingly hot. Additionally, there's a lot of pressure on the splines and a lot of back and forth sliding motion. Any liquid type of grease will quickly be squeezed out of the splines, and leave you with no protection. The first time you remove your rear tire, you will likely find that the factory grease has dried out completely and solidified into something which does not even remotely resemble a lubricant. Actually, if it's a good Moly grease, it's still doing its job in this form, but it's not a pretty sight.
Honda specifies a spline grease which is 60% molybdenum disulfide ("moly"). Moly is a dry lubricant which bonds to the metal surfaces, offering lubrication properties even when the parts have squeezed everything liquid out. A lot of greases now say "Moly" on the container, but you must be careful about this. Ford and Caterpillar specify moly greases for particular applications, but the requirement is for 3% moly, not even close to the 60% requirement of Honda. It's not enough to buy a moly grease, what you really want is basically dry moly in a grease-like carrier which makes it easier to apply.
These days, most BMW shops seem to be using the Honda moly paste on drive splines. I have no idea what Kawasaki, Yamaha, or Suzuki dealers are doing, but if it's anything like what most Honda dealers are doing, it's simply not acceptable. My informal survey of Honda shops, backed up by observations from several other riders, has convinced me that essentially none of them use Moly-60. They use the cheapest brake and drive shaft grease they can buy.
If you let a dealer or shop change your rear tire, be certain they are using Moly-60 paste or Krytox, or you're going to be needing new drive splines in about 50,000 miles. Guaranteed. I recommend you remove and replace your own wheels, leaving the spline cleaning and lubrication up to you. If you want nothing to do with this, then I recommend you have a talk with your favorite mechanic before tires come up, and buy your own tube of lubricant if necessary.
Moly greases with 60% + molybdenum disulfide content:
An alternative to moly is the new poly-flourinated lubricants made by DuPont called Krytox Teflon Bearing Grease. These chemicals are simply magic. They have almost no known solvents, are chemically inert, and don't burn at any temperature, even in a pure oxygen atmosphere. This is pretty clearly the only grease to use. See this article. Nascar mechanics have found that Krytox grease can reduce the temperature of spline joints on drive shafts by 150°. Also, this stuff lasts forever. It is, unfortunately, quite expensive.
Krytox is compatible with moly - in fact DuPont sells a high pressure Krytox which mixes the two. You may hear some horror story from some mechanic about mixing greases and the result turning radioactive or some such. Don't worry, it won't happen with moly or Krytox.
Hint: This is a quite messy job. If you get some spray shaving cream and rub it into your hands and wrists, it will seal your skin and pores with silicon lubricant. Later, when you're all done, this will make cleaning up your hands a lot easier.
With your rear wheel off the bike, the drive splines will be exposed. They're the brass colored bumps in the middle of the drive ring, above right. Clean off any factory stuff with a rag. You will likely need some solvent like Kerosene or Gasoline to get them clean. Now put Moly-60 paste or Krytox on the exposed splines. The idea here is to more or less paint the splines - you're not looking for a large volume of grease, like you would use on a wheel bearing. You want an amount of grease about the size of a pea. It's helpful to use a clean popsicle stick to spread the grease. Try not to get too much on your hands, neither the Krytox nor the Moly greases clean off very easily.
At the first tire change, you should also remove the rear drive system from the swing arm and clean and lubricate the splines on the drive shaft. After the first time, this should be done about every 50,000 miles or so.