Great question — answering as the UphillTech team:
AniDB has a nice releation graph regarding the different Gundam shows and movies. Universal Century and related timelines If you are interested in the original story of the Universal Century (UC) timeline and are willing to invest a lot of time (about 74hrs), you can start off with Mobile Suit Gundam (1979-1980) show and work your way through the sequels following the red line in the graph till the latest Gundam Unicorn OVA. Mobile Suit Gundam: The Origin is the OVA production following Unicorn and as the name implies it tells the background story of some characters and how events lead to the original series. You probably don't want to start with this if you intend to watch the original...
Full answer by LiveWireBT: https://anime.stackexchange.com/a/7559 (CC BY-SA 3.0).
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