People sometimes ask, "How do I start watching Gundam?"
It's simple: watch Mobile Suit Gundam.
The first Mobile Suit Gundam is a 1979 animated television series. It was later adapted into a film trilogy. Both versions are excellent. This original story (in either form) is the only thing you really need to see in order to understand where later Gundam is coming from. After experiencing the original Mobile Suit Gundam, you can enjoy the Gundam franchise in any order. Each entry tells its own story. Just watch what interests you, and skip what doesn't.
"Does Gundam have an ongoing continuity between entries?" Sometimes. Mobile Suit Gundam (1979) is set in the future of our world, in an era known as the Universal Century. This setting has been revisited and expanded by many later Gundam works, and is the focus of this fansite. Even within the UC, different writers take their own liberties with the canon, so various interpretations of the setting co-exist. There have also been a number of more self-contained Gundam stories with their own continuities, beginning with the Future Century of Mobile Fighter G Gundam (1994).
"Okay, I watched Mobile Suit Gundam. If I want to see the rest of this storyline, what should I watch next?" For the "classic 1980's Gundam" experience, watch Zeta Gundam followed by Char's Counterattack. Like Mobile Suit Gundam, Zeta is a television series that was later adapted into a film trilogy. The original television run of Zeta is a stone classic, so I recommend it. The films are also good, though, and manage to be a bit less depressing and sexist.
After Zeta comes ZZ (called Double Zeta), which you might want to skip. There's some good stuff in it, but it is early Gundam's low point in terms of tone, plot, and "anime bullshit".
Finally, Char's Counterattack is the grand finale of the story so far.
"What if I want to watch all of Gundam?" If you are determined to witness the complete franchise, then you might as well watch everything in production order, the way the original audience did.
"Should I watch Gundam in chronological order?" Nope. Mobile Suit Gundam: The Origin (2015 OVA) is a prequel based on flashback chapters from Yoshikazu Yasuhiko's manga adaptation of the original series. Watching Origin first would needlessly spoil a number of plot twists (and iconic character introductions) in the original series. Furthermore, without the context of the original series, Origin loses its dramatic irony.
"Okay, so I can watch Gundam in pretty much any order. But what is the best order to watch Gundam?" If you want to experience the entire* Universal Century in the most narratively satisfying order, consider my (very opinionated) recommended viewing sequence below.