Reddit mentions of Mobile Suit Gundam Movie: Trilogy DVD Set

Personally, I grew up with Gundam Wing on Toonami so that will always hold a special place for me as a starting point into the Gundam Universe. That being said, it's technically an alternate universe to the original "universal century" timeline. Starting with the original Mobile Suit Gundam would be the chronologically correct way, but it aired in 1979 so the animation is a little... well I'll just say old school. They did remaster and condense it into 3 movies which is nice so you could start there. It's ~$30 on Amazon

If the animation seems too old school for you, I would recommend The 08th MS Team which takes place during the original timeline but is a side story. They also aired it on Toonami when I was growing up so it was my entry into the main timeline of the Gundam universe. I bought it on Amazon back in the day but it seems to be unavailable now. If you Google "stream 08th ms team" the first link seemed to work.

Otherwise, if you're okay starting with an alternate universe setting, start with Gundam Wing like most of my generation did. The mobile suit designs are great and you could compare some of them to Titanfall in that each has their own uniqueness to them. Heavyarms = Legion mixed with Tone, Wing Zero = Northstar mixed with Ion, Sandrock = Ronin (if Ronin had an SMG instead of a shotgun). Anyway, I also bought the complete collection on Amazon back in the day but it's sadly unavailable now as well. I would say just torrent this one.

You can find one in 'The usual places' if you go that route, otherwise if you'd like to buy it Here it is on Amazon!

I'll warn you, the original english audio is...terrible. Char is pronounces 'Sha' and just...doesn't have as much going for it as the English Series dub, as it was done by a different company. Still, if you want to see the original MSG in a shorter form, with some bumped up animation at times this is the way to do it.

Ninja Edit: The Amazon version I linked apparently has the updated Japanese Sound-Set. While this sounds amazing, it may seem a bit 'off' from what you expected if you had, say, seen the TV show or even a copy of the original theatrical release. As I said though, it's pretty fantastic. Not sure if it's got the English Dub either...seem's subbed for sure however.