Do you know the best current method I can use to root my s4?
Hi, welcome.
First of all, keeping the stock rom is a great idea if you don't want to deal with flashing custom roms. I have my S4 rooted and using the stock rom. I use Xposed modules (specifically one called Wanam Xposed) to theme Touchwiz the way I want it to look.
I have a Verizon S4 running 4.3. I used a method called Safe Root in order to root it. Search for it online. However, you must make sure that you are rooting with the proper method by doing a search on (for example) "how to root [carrier] galaxy s4 [build i]".
Also, your bootloader is most likely locked. If you try to install a custom recovery you could soft brick your phone. If this is the case you can use a program called SafeStrap to flash some roms.
In any case, do more research about your specific phone (model, carrier, build, software version) in order to find a suitable root method. Good luck! Oh yeah, if you root then take an OTA update you will lose root and depending on your phone you may not be able to get it back