Welcome to the forums.
First off you need to root your device in order to install a ROM. After that, I would make a Nandroid backup in case something doesn't go quite right, then I would flash your ROM.
Edit: I hit post too soon. Read over the rooting instructions several times. Any questions, I can try to help as much as I can. (Flashing ROMs has changed a little since I had my S3)
I've run a couple of different ROMs on my Note 3. IMO, none are easy to do in the way that a Nexus device is easy.
I've stuck with Touchwiz based ROMs for the S Pen support mostly. The main hassle with Touchwiz ROMs is keeping up with the official firmware update tweaks to work with the latest ROMs. Where I've been rooted and ROMmed since 2013, this usually means flashing the latest stock factory ROM and bootloader bits via ODIN. Re-rooting the phone, Flashing the latest Recovery (TWRP or CWM)--more ODIN. And then the chosen ROM which might be via recovery or OdIn depending on the ROM.
First I would ask what you want in a ROM that you don't get in the stock one. There's no reason to just flash a ROM. It's like having your car painted when the current paint job is fine.
Most of what most people want a ROM for they can get with a launcher, a theme and some wallpaper.