I'm no expert either. I've been reading on xda. One important thing - if this phone which is purchased or OTA-updated to recent versions (MF3 and MK2) then it has Enhanced Security Linux with locked bootloader, so flashing is not so easy.
To the best my knowledge, flashing a custom rom with Odin is not an option and you could brick your phone trying. (there is an odin back to stock on xda).
In fact, there is only one way to flash and that is using a program called safestrap
I think you are on AT&T, correct?
The program page for safestrap for S4 is here:
http://forum.xda-developers.com/showthread.php?t=2448925
The Q&A page for AT&T S4 phones is here
Official AT&T S4 Safestrap Discussion Q/A - xda-developers
It's a lot to read.
Even with safestrap, the ROM can be changed but the Kernel cannot be changed. That means you are limited to ROMs that use the same kernel as the one you have now. The kernal you have now is Touchwiz on Android 4.3, so you can only use custom Roms that have that same kernel. Last I checked, there were few or no Roms available in that category (there were a lot more custom roms that use Touchwiz / 4.2.2 kernal).
You have to comb xda to find ones that might work. I haven't seen much of a list. Here's a few:
Mention of ROMs that might work: Goldeneye:
http://forum.xda-developers.com/showthread.php?p=49528405&highlight=4+3#post49528405
Deadly Venom and Dark Carnage (sounds gruesome)
http://forum.xda-developers.com/showthread.php?p=49435267&highlight=4+3#post49435267
You might stumble accross a few others but there's not many.
I haven't installed safestrap yet, still looking at the options. Safestrap can require a lot of room on your internal sd to house the stock rom and additional roms.
The part about S4 Google Edition Rom - I mentioned because you said you wanted google experience. I remember reading about G.E. rom somewhere (added link below) and that's why I mentioned it. But now I'm thinking it won't work. For one thing it needs to be a Touchwiz kernel (it's probably AOSP so wouldn't work), also not sure if 4.3. It mentions Loki which applies to older phone firmware than ours (which use safestrap). You're welcome to look into it more but I'm thinking it won't work:
http://forum.xda-developers.com/showthread.php?t=2365317