Do I root to my model, or my version number? When I rooted my last phone, I was able to find a detailed tutorial made specifically for the model I had. It was great, because I knew I had fully compatible tools and ROMs, which limited my amount of fumbling around.

I don't see anything out there for my new (to me) phone. I've seen generic guides for my Android version, but since it's specific to my carrier, I am wondering if I can just jump right in and start trying it out.
In order to root an Android phone, you have to take into account the Manufacturer of the phone and the Model, and also you need to take into account the Android Version installed. The process to root a phone is based on some exploit in the system, and each Android version with each model has a different method, which takes advantage of different exploits.
There are common exploits across some models, but in general, you need to look for a rooting method for your phone model and the Android Version it has.
I have used this page on some occasions to look for tutorials, but you can also look in xda developers