After years of helping new rooters, this is my advice to those considering rooting for the first time...
First, you need to determine if you must root the phone to accomplish a particular thing you need or want to do. Or, if not then maybe you just want to experiment. Nothing wrong with that, but there's some things you should know and consider first to avoid unnecessary grief.
Rooting grants full administrator permissions to the user, rather than limited user permissions. The good is that this allows you to do basically anything to any files on the device. It allows full control of everything on the device. The bad is that it also means you have full permission to muck up anything on the device. It also reduces the security of the phone somewhat.
I used to root every phone I had. But I no longer root my daily driver phone that I cannot do without. As hardware and software has improved the need to root has declined. Now I do not recommend rooting unless you have a specific need or just want to experiment on a spare device.
If you decide you really do need and want to root, first do a LOT of reading. See if there's a subforum here for your phone. You should check out the XDA Developer forum for information too, but be sure to read first and post later there as it's not a very n00b-friendly site. And as always, for questions this forum and Google search are your friends.
There is some chance a mistake could 'brick' your phone (make it unusable). Make sure you understand why you're rooting, the exact procedures required and all the terminology. And it's absolutely critical to have a recovery plan and backups in case things go bad.
DO NOT try to install a ROM that is intended for any phone except your exact model! If you have questions post back here before proceeding and someone will help.
Don't forget to do backups! Read about nandroid backup.
Until you are confident that you have everything above ready and understood, you're not yet ready to root.
Android since v1.0. Linux user since 2001.