Gundam Wing - This series just oozes 90s anime style and cringe in a way that can get a little annoying as an adult, but it's a genuinely good show overall. It's a good war story with good action and decent political drama, but it's focused through this lens of broodiness that may not be everyone's cup of tea. And be careful not to cut yourself on all of that 90s edge!
The Origin - This is a prequel story to the original Gundam series that focuses on the story of one of the main antagonists, Char. It's a fun story with good animation and action as well as pretty nice political drama, but it falls short with a lot of people because this Char is shown quite a bit differently than the Char from the original Gundam series onward. I wouldn't recommend this as a starting point, but I would still recommend still watching this one at some point after seeing the original Gundam series.
Iron-Blooded Orphans- This one is a fairly modern series set within a pretty brutal world. If you're looking more for a "horrors of war" kind of story, then this is probably the one I'd recommend. It's definitely not a very happy story. The first season is fantastic, but the second season is only "pretty good" in my opinion since it never quite hits the same stride.
Gundam 00 - A fantastic series that I can't recommend enough! If you only want to watch one of the longer Gundam series, this is the one I'd pick for newcomers. It has great action sequences, the animation is still good enough to not feel dated, and it has a good mix of drama and character development! There is also a movie set after the series called A Wakening of the Trailblazer that is generally seen as a bit of a black sheep, but I'd still recommend watching the movie if you enjoyed 00.
08th MS Team - This one takes place during the middle of the original series, but it is a stand alone story that requires very little knowledge of the original series to appreciate. I tend to find this one to be more of a love story set in the middle of a war rather than just a straight up war story though, so I think it's worth keeping that in mind. If you are looking for a short series to wet your whistle and dip a toe into the Gundam universe (specifically the Universal Century timeline), this is a great place to start. And it's a fantastic series to boot!
Unicorn - Unicorn is what got me back into Gundam as an adult. It has great animation, but while I think you can certainly start here if you'd like, Unicorn is very much intended to be a sequel series. As such, you would likely get a much better understanding of the details if you watch the previous entries in the UC first, but I think you can still follow along with the main story of Unicorn just fine if you just want to jump right in.
Seed Freedom - I haven't watched this movie yet, but it's a sequel to Seed Destiny (itself a sequel to the original Seed). I'd recommend holding off on this one until you've watched Seed and Seed Destiny if you can. The Seed stories have a bit of melodrama that was typical of anime series at the time, but they're both very good. And don't let the naysayers dissuade you from at least looking into Destiny! It's still an above average story with great action, but it just falls a bit short of being as good as Seed was (because Seed itself is excellent).