Well, a ton of shows were influenced in a ton of ways by Gundam. It would probably not be a stretch to say that every (or almost every) robot show of the 80s and 90s owes something to Gundam, so you'd kind of have to get a little more specific about what you're looking for.
Several shows have already been mentioned. Macross and Patlabor are both fantastic. Both are members of the "real robot" genre that was started by Gundam, but they're otherwise rather different from Gundam. Still, they're both very good.
The ties between shows like Escaflowne or Evangelion and Gundam are a little more tenuous, but they're there, and Escaflowne at least is a wonderful show. It's been a while since I watched it, but I would say Escaflowne's closest tie to Gundam is the fact that both are robot shows featuring a human drama set against a backdrop of warfare; prior to Gundam, most robot show antagonists weren't human, they were things like a mysterious alien force or what have you, so any robot show that features human vs human warfare has that, if nothing else, with which to credit Gundam. It's also been quite a while since I saw Evangelion, but if there's nothing else, there is at least one thing that ties it to Gundam. I recall a while back, stumbling upon a website that compared Shinji with the main character of an unnamed robot show. He spent the entire time making all sorts of connections between Shinji and the mystery man and complaining that the creator of the other show had obviously just ripped off Evangelion. At the end, he revealed that he was actually talking about Amuro Ray of First Gundam. He wasn't a moron who didn't realize that Gundam was sixteen years Eva's senior; rather, he was trying to show all the militant Eva fans that even Hideaki Anno is not a god among men, that his ideas came from somewhere, and that just because Eva did it doesn't mean that Eva did it first or that Eva did it best. Now, I can't say with certainty that Anno specifically had Amuro in mind as he created Shinji's character (well, the parts of Shinji's character that weren't based on himself, that is), but there definitely are a number of parellels there, and the people at Gainax make no effort to hide the fact that they're Gundam fans, so...I don't think it's too great a leap.
Lots of other Yoshiyuki Tomino shows also share similarities with Gundam. I've actually never gotten to see any of his pre-Gundam work, so I can throw out some names, but I can't speak for whether they're what you're looking for. Some of his post-Gundam shows include Ideon, Xabungle, Dunbine, and L-Gaim in the 80s, and more recently, Overman King Gainer (just to name a few). I regret that I have been utterly unable to get ahold of Ideon, although I desperately want to see it, so I can't speak for it. The same goes for Xabungle, except that I'm not really as adamant about seeing it. I have had the opportunity to see parts of Dunbine and L-Gaim, but not much of either; what I've seen is pretty good, something I would recommend, but I can't say it doesn't get worse later on (I will say that that would run counter to everything I've ever heard about either of them, though). Just based on what I've heard about it, I think I would probably recommend Ideon, but since I don't have firsthand experience with the show, I'm hesitant to tell you to try to get your hands on it. Overman King Gainer, I'm happy to say, I have seen in its entirety. You'll find that Tomino's style and his themes have changed a great deal over the past twenty years, though. King Gainer shares a striking similarity with a certain Gundam series, but it's not First Gundam; rather, it has a lot of ties to Tomino's latest (and one of his finest) Gundam productions, 1999's Turn A Gundam, which was drastically different from any of his previous Gundam outings. If you're looking for shows similar to First Gundam specifically, then King Gainer isn't what you want. If you're looking for shows similar to any member of the Gundam franchise, though, then it's as good as any.
To cover your other question, yes, the movie trilogy is a fine way to start. Personally, I like the trilogy more than the TV series, but that's just me. The Gundam franchise is riddled with some pretty terrible movie compilations, but the First Gundam compilations are not among them. They're very well done, really a perfect compression of the TV series. I would go so far as to say that if you see the movie trilogy, you don't even need to bother seeing the TV series unless you just want to see everything for the sake of completion. The one thing I will warn you about is that, yes, First Gundam was created in 1979 and its age shows. The video was completely remastered, so the colors are pretty vibrant and the picture quality is good, but the animation itself is dated and the character and mecha designs are definitely anything but modern. To me, none of that is necessarily bad. I like the mecha designs, even though they're relatively simple; they're either good, or they're cheesy in an enjoyable way. I also really like the character designs, but obviously they're based on what was popular and fashionable in the 70s, not now, so they also look dated in that way. If that bothers you, there's really no way around it. Furthermore, the movie trilogy is mostly made up of animation taken directly from the TV series, so it's not a great leap forward in animation quality. The second and third movies both feature a good deal of new animation, but it keeps a pretty consistent look. So regardless of whether you watch the movies or TV series, you're going to have to deal with the dated look. If you can live with that, you're in for a great story and some great characters.
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