Hi I have a few questions about using a phone made for another neterwork (att) on tmo

Well in short I have an At&t Model GS4 the i337 its currently unlocked running on stock Nj4 and of course want to root it. First of all i suspect i need to use t mobile roms now that i am on that network.
Here is a list of questions and links are also appreciated as answers anything that helps!

First and foremost how should i root this (my last device was a G3) so if things are different please tell me also, I their an app or a thread you can point me to.
Secondly how do i update it to a v5.1 or the latest so updating them rom later is not a problem if i must use odin to do this (i want to use the latest to use odin the least)

Lastly anyone who can give directions on how to root and get to the latest version of android i wold greatly appreciate the time anyone spend to respond to this this post i TYIA

So any links that will work work with NJ4 I would be so happy I have only experience with the G3 I feel i know what to do with it but the Gs4 is new to me and i appreciate any help i receive/

Thank yo for reading this and TYIA for responding thank you for your help i hope its as simple as someone posting some links so i can do this easy i again thanks for leading me through the fog and TIA