First, I am not an opponent of root at all. I have always rooted all of my devices for many reasons, most of which have already been listed. I do want to point out that for most people root is not needed. You can tether with a USB cable using ClockworkMod Tether - https://market.android.com/details?id=com.koushikdutta.tether. I recommend thoroughly reading as much as you can about the root process before you attempt it. It is fairly easy to do, but it does require you follow the steps carefully and it is even better if you understand what you are doing as you go. Screenshots can be done with the Android SDK and a computer as well, but I have never used that method.
Titanium Backup is a very good app to have. It requires root but you can backup all your apps, even at scheduled intervals, and restore them as needed. It is great if you try custom ROMs.
There are some good reasons to do it but know the risks of doing it. It is possible in certain steps of the root process or (less so, but still possible in the) flashing process to permanently damage the phone. It is fairly difficult to do if following the guides to the letter but it is possible if you miss a step or something doesn't match (like an MD5). That will not be covered under warranty if something happens. Again, this is not meant to discourage you, but merely to make sure you are aware of what you are getting into before you do it.