I have a samsung galaxy s4 (at&t) that has just gotten the android 4.4.4 update. I have wanted to root my phone for a while but is their even a way to root it safely? How would I go about doing it.
I would give this site a thorough read before trying it. http://www.xda-developers.com/tag/all-android/. Also if you don't mind losing your data or backing it up and starting over, flashing your phone to a variant of android that already contains the binaries/busybox would be your best bet. With the latter method however you will be doing low level work with your NAND flash and the chance of bricking your phone is about 50/50 if you suddenly lose power in the middle of a push. So be careful and take it slow; five minutes of your time saved isn't worth the $400-800 to replace your phone.
Galaxy S4 user here. Rooted my phone last week using Toweroot. Worked for me.