First, you can choose your Verizon number, if you wish.
Google has plenty of info online about GV.
Getting Started Guide - Google Voice Help - getting started, tells you what each option has
Using a Google number vs. a non-Google number : Basics - Google Voice Help - this should be helpful.
About Google Voice : Basics - Google Voice Help - good start for GV with a Google number
Using a non-Google Number - Google Voice Help - complete info about using a non-Google number
I'd suggest using a Google number, but all of that info above should help you decide for yourself.
Ok so i looked all over the google voice page on google and I have one question. If I add google voice to my phone will it replace my phone number verizon gave me?
Now, it does not. You can add any of your phone numbers to link to your Google Voice number. I have both a landline phone at home, and my VZW phone linked to my (different) GV number. By default, if somebody calls your GV number it rings all of the phones that are tied to the account, and the first to answer gets the call (you can do any number of customizations, which I will get to later.) When you place a call with Google Voice from the web site, it will ring the call, then (by default) ring all of the numbers linked to GV and, again, the first one that answers gets the call. Obviously, if you link only your VZW number to GV, it uses just that number.
I believe that if you place a call with the GV Android app, it just links to your VZW number by default.
As for customizations, you can do things like always direct calls from a specific contact group to a specific phone number when your GV number is called, I believe that you can schedule all calls to go to a specific number by time of day. You can direct a specific group to go straight to voice mail. There are a number of things that you can do.
or how will I have the option to use my verizon number or my google voice number when calling people?
Yes, you can set it up to always use your GV number, always use your VZW number, let you choose for each call that you make, or use GV only for international calls.
I'd strongly suggest getting a Google number. It gives you a lot of flexibility - if you are married, you can have both of your phone tied to the one GV number, if you wish, and by contact group as well, so you can have your Mom's calls always go to your phone, etc.