[Galaxy S III Q&A, Help & Troubleshooting] Timusius:
I was extremely new just a few weeks back as well. I must say that my attempts on finding a solution for my slow S3 has been a rather hard and bumpy road.

To help you save some time, here is what I've gathered so far:

Last time I flashed phone firmware, a couple of years ago, it required a PC, a cable and some funky software.

Then when I flashed my S3 for the first time, about a year ago, I started out with Cyanogenmod and used their app+cable installer. I think this is still the easiest way to root the phone, and get started with custom roms. Even if you plan to use some other rom than Cyanogenmod.

When the phone is rooted you can install TWRP, simply done via an app on the appstore. TWRP is recovery firmware that can stay on the phone, even though you flash new roms. (The phone is divided into several separate sections, bootloader, recovery, actual system, and data area.... I don't even know much about it yet.) The important part is that it is extremely easy to flash new roms using this tool, and a micro SD-card.

Simply drop the rom zip file onto the card and flash them. (For some roms you need to flash some "GAPPS" as well. These are the google apps, like the play store etc. If you don't flash them you can't really use the phone for much.)

This is where the real problems start. Finding the right ROM.

The S3 is old, and only supported up to Jelly Bean for my 1GB international version, or Kitkat if you have one of the 2gb versions.

This has not stopped developers in providing both Kitkat and Lollipop to any version of the S3. However... as eager they are to provide the newest coolest features, just as lazy do they seem to be with their testing. (Sorry guys, but someone has to say it.)

So I've gone through some roms now... none of them really good.

The first Official Cyanogenmod I flashed half a year ago. I used this for almost a year.. but the camera would crash constantly, and the phone was not able to display video. For example movieclips on facebook.

Second ROM: Cyanogenmod has also stopped developing for the S3, so a few weeks back I switched to some unofficial version. It seemed to work... camera and video was ok, and everything had the nice Lollipop feel to it.

However MMS did not work. I could receive one and send one, next ones would require a reboot for them to work. The problem was brought up in the ROM Q/A tread, but the answer from the developer was: Sorry I don't use MMS so I don't really see a problem.

Third: Onto some next rom. This one was tagged STABLE. It was far from it. In fact it was even easier to discover that this rom had the same MMS problem. Because the message app would crash immediately when trying to send one.

4th attemt was the same rom, but a different kernel... still the same problem.

5th rom was Blisspop... Wau, this was beautiful! Cool webpage, Cool animations.... it was just too bad that the site is dead, and that the rom can't even run for 24 hours without the android keyboard crashing 25 times or so. Also Blisspop has OTA updates... the only problem is that it constantly says that there is a new version for the S3.. but the download fails. Apparently some bad change to the website caused this, and apparently this problem has been going on for 3 months without anyone bothering to fix it.

6th was Archidroid. The second to top ROM for the S3. It looks cool, and has many features... but just as other Lollipop ROMs, sending MMS's without error is not one of them....

I've really tried everything. People always says that the MMS problems are due to APN's not being set up correctly, doing some reboots etc.... but trust me... I've set them up so many times you wouldn't belive it.

7th. After reading this thread and rp158's recommendation of NCS-ROM I'm now running that one. Without a single thing being set up, I was able to send and receive MMS's right away.

However... now I'm stuck back at the ugly kitkat look, and Touch wiz.

tl;dr;

So what do I want to say with all this?

Get root and TWRP and start playing around with ROMs :)

Lollipop does not look like an option for the S3, at least not right now :(

Don't let yourself be fooled by nice webpages or [STABLE]-tags, or comments that say: "I've been running this since yesterday.. it's 100% stable!"