Each has its ups and downs.
One of motorcycling's frequently asked questions is which of the three common drivetrains—chain, belt, or shaft drive—is best. You might as well ask what type of oil is best, or what tire you should use. Opinions are like the exhaust outlet of a donkey because everybody has one. Still, each type of drivetrain has quantifiable advantages and disadvantages. Cyclecruza summed them up well in this video, and I'll add my two cents' worth as well.
Arguably the most common drivetrain, chains are most commonly used in sportbikes, dirt bikes, and some cruisers and touring bikes. A chain drive is the most efficient way to transfer power from the engine to the back wheel, with only 3 percent of the power lost in translation. They're strong, which is why they're good for drag racing, and they're durable, which is why they're used almost exclusively for dirt-oriented motorcycles.
The biggest disadvantage of chains is that they're high maintenance. You need to clean and lube the chain frequently. When you replace the chain, you also need to replace the sprockets, adding to the expense and labor involved. A chain drive is not as smooth or quiet as a belt or shaft. Plus, you need to carefully align the back wheel so that the chain rolls on and off the rear sprocket straight.
Belts work very much the same way as a chain. They're most common in cruisers, because they're quiet, smooth, and low maintenance, with no lubrication necessary. While you need to lube your chain every few hundred miles, belts can last for 15,000 to 20,000 miles, or maybe more with expensive kevlar belts. Their power loss is about 11 percent, which is more than a chain but still not bad.
Despite their longer life, belts are more prone to damage than chains. A pebble stuck in a sprocket can wear through a belt and cause it to fail, something that will never happen with a chain. It may be possible to temporarily splice a broken chain back together on the side of the road to get home, but there's no way that's happening with a belt. A replacement belt is more expensive than a chain, but you can usually reuse your old sprockets, which helps offset that cost.
The ultimate low-maintenance drivetrain is shaft drive. The only regular maintenance required is to change its fluid from time to time. No wheel alignment is ever needed because there's nothing to align once everything is bolted up. Aside from catastrophic failure, a shaft drive will last the life of the motorcycle. These traits make it popular on touring bikes, as well as some cruisers.
Because power has to travel through so many moving parts to get to your back wheel, 31 percent of that power gets lost on the way. That means that on my Honda PC800, only 38 of its 55 horsepower actually got to the road. On top of that, shaft drive adds a great deal of unsprung weight, which hurts handling. No wonder that bike felt slow.
The other major disadvantage of shaft drive is torque rise. The force of the driveshaft turning the back wheel causes the rear of the bike to lift under hard acceleration, rather than hunker down as you'd expect. This prevents you from taking advantage of the improved rear traction that rearward weight transfer provides. If you're touring the country on a Gold Wing you probably don't care, but this is a huge disadvantage for a sportbike.
So Which Is Best?
The definitive answer is "it depends." I wouldn't want anything besides a chain on my Kawasaki KLR 650. I ride dirt, which is bad for a belt, and I don't want a big shaft drive adding to its already significant weight. Yet I loved my PC800's shaft drive for its low maintenance. A belt was the best choice for the Suzuki Savage I had, both for its low maintenance as well as its ability to transfer more of its limited power to the back wheel than a shaft.
So the true answer is that it depends on the type of riding you'll be doing and the specific motorcycle you plan to do it on. Belts and shafts are both great for comfortable touring, but if you want a cruiser, you'll probably end up with a belt, where a sport-tourer would most likely have a shaft. For performance, either on-road or off-road, a chain is your best bet despite the extra maintenance required.