Are there any benefits/downsides to installing an Xbox 360 game to my hard drive before playing it?
One of the benefits would be that your disc will get less wear and tear. You still need to put the disc in, but after verifying it, it shouldn't be read again (though that will depend on the game). This should minimize the danger of scratching it if the console gets knocked. There should also be less noise as it doesn't need to keep the optical disk spinning.
You might also get faster loading times - but that would depend on the game.
An obvious downside is that you are going to fill your hard drive if you do this for all your games.
In general, you will get faster loading times reading from the hard drive than the disc. However, this is not always the case, especially with some of the older games.
One example is Halo 3. Don't install Halo 3 to your hard drive. It will actually perform worse than if it read off the disc (because the game was already made with caching from disc to hard drive). The reason for this behavior is because Halo 3 was released before installing games to the hard drive feature existed and therefore they programmed their own caching to the hard drive as a part of the game.
Looks like installing might allow you to play discs that are too damaged to play directly, which is a huge advantage in the right circumstances.
Playing on the "con" side of things, remember that while playing from the hard drive will reduce the wear and tear on your Xbox Disk, it will increase the wear and tear on your hard drive. If you play all your games from the hard drive, this will absolutely reduce the life span of it. If you have your game saves on your hard drive, then imagine what would happen if your hard drive crashed. Yea, start over on everything not saved in the cloud by Xbox Live. A way around this is to keep game saves on a memory card, but that's not a lot of room.
So, not necessarily a deal-breaker, but definitely something to keep in mind. Your hard drive will likely die long before your optical disk/drive will.
Another benefit that hasn't been mentioned yet is that in some games, you may be able to see/hear more things.
This is because some games only load mandatory/necessary video and audio from disc while loading all available content from a hard drive, as hard drive reading speeds are considerably faster than disc reading speeds.
I'm not quite sure how many games do this. However, I saw a mention of this recently by Bungie, the awesome company behind the Halo series. This means that Halo: Reach might employ such a strategy for balancing read/load speeds.
For a good list of load time improvements look at this post on the Cheap Ass Gamer Forums. The first post, click the "show spoilers" button.
Looking at the post again, it refers to GameInstalls.com where the install size and speed increases are reported by users.
EDIT: Looks like the Cheap Ass Gamer post has some better data for a few games than the GameInstalls site.