Ok… there we go…
Perhaps the single best Gundam full series to recommend is Iron Blooded Orphans . It’s modern, it’s snappy, and it has all the requisite stuff.
But…
It belongs to it’s own timeline, and I believe to properly appreciate Gundam it would be good to watch stuff from the main Timeline, the UC, or Universal Century. It’s also the best series to recommend because it’s modern, but it’s not the best overall, imho, unless you can’t really stand older stuff.
The problem with the UC is that by far the best works are not the main series (the original, Z and ZZ), but OVA’s and more limited series (in particular, War in the Pocket, the 8th MS Team and Stardust Memory) which you will enjoy more if you know a little about the One Year War (the original series narrative). You can watch them on their own and they will hold up. War in the Pocket is specially interesting, I think.
There’s also the pretty amazing OVA Gundam: the Origin, but it’s only amazing if you have seen the original series and Z Gundam. It’s sort of an origin story for the bad guy and it’s great if you know the mythos. Otherwise it will feel slow.
The original series I enjoyed, but it’s really, really old and sort of hard to watch unless you put rose tinted glasses on. You can sort of watch it on the background while doing other stuff. It does lay down a lot of stuff that is used in other series, though. Z Gundam is also old, but much, much better than the original (it’s really good for when it aired) and much more watchable. However, it does not finish it’s story. It finishes in ZZ Gundam and that one is a little worse.
Also UC but so far removed than it can be watched on it’s own is Gundam Unicorn. It’s pretty good, quite modern, and has great mecha design, but it felt too much like a rehash of a lot of stuff to me. But it might land better on somebody less familiar with the previous works. But for a straight up series I would still go with Iron Blooded Orphans first.
So:
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If you want to see a single series: Iron blooded Orphans. You should watch this anyway.
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If you want a more fulsome classic Gundam experience perhaps War in the Pocket and the 8th MS team, and if you really enjoy those then the original series, followed by Stardust Memories and then Z & ZZ Gundam. After that either Char’s counterattack and Unicorn or Gundam: the Origin, depending if you want to see how things evolve (Unicorn) or you want even more insight in how the One Year War and Char came to be (Gundam: the Origin).
Yeah, it’s a mess. I did do the chronological thing at first.