S3 ROMs don't work on S4 models and vice versa.
So you found the ROM you want to try, so go ahead and download it.
Connect your USB cable to your phone. View the folders on your phone. I like to make a folder called ROMs on my phone and transfer it in there.
After you transfer it, open my toolkit and press Boot into Recovery. When you are in TWRP, MAKE A BACKUP!
After you make a backup AKA a Nandroid, click Wipe. Wipe Cache, Dalvik, Factory Reset and System. NOTHING ELSE! If you click anything else, IT WILL WIPE YOUR SD CARD AND YOU WILL HAVE MORE ISSUES ON YOUR HEAD!
So you wiped everything. Great. Your phone is ROMless, Yay!
Now go back and press install. Go to the folder where the ROM you downloaded, is located in. Select the ROM and slide the screen to flash it. After it flashes, press the back button. Press Reboot and Reboot into bootloader. Open the ROM you downloaded from your computer. You should see boot.img. Extract it to your computer. Open the Toolkit and press Flash a Kernel. Locate the boot.img and it will flash it.
After it flashes, press reboot from the phone.
Congrats! Enjoy your fully working or borked ROM!