The Gundam series is home to 7 different 'Centuries' or, universes: The Universal Century, The Cosmic Century, The After Colony Century, The After War Century, The Future Century, and The Anno Domini Century. Most centuries are not linked together, though I'll explain the one exception later.
The Universal Century is the first of these 'Centuries' and includes the original 1979 series, the two sequels series, the two miniseries, The live action show, the movie: Char's Counterattack, and The Gundam F91 film. It starts off a little slow, and the original 1979 series has some animation quirks that might turn you off, but it's considered the best 'Century'. However, all of these series are unlicensed right now, so you'll have to go to an anime streaming site to see these. RightStuf might be bringing out the original 1979 series and it's two sequels.
The After Colony Century was made after Tomino, the creator of Gundam, basically went crazy during the production of Victory Gundam, and new people were brought in while he took a break. The AC is only home to one series, Gundam Wing. It was on Toonami a long time ago, and is the most popular Gundam in the US. It's considered subpar by most, though it did get a sequel, Endless Waltz.
After the AC's failure, Tomino came back and produced Turn A Gundam, the only series in the Correct Century. It's currently the only Gundam readily available in North America.I haven't finished it yet, but I understand it is the end of the Gundam franchise.
The Cosmic Era is home to the Gundam SEED franchise, and has a mixed reception. Most agree it was great for the first half.
The After War Century holds Gundam X. It's the only Gundam series to be cancelled in Japan, and one of the only animes to be cancelled.
The Future Century, again, only has one show: G Gundam. G Gundam is a lot different than any other Gundam out there. It's a lot more simple, and has a lot more battles. It was made for a younger audience, but many find it to be, if anything else, entertaining.
The Final Century we'll talk about here is the Anno Domini Century, which is the Century for Gundam 00 and it's film. I wasn't a fan of 00, and it seems to be generally ignored.
If I had to recommend you one show on here, It'd be the original 1979 series. If you can get past the animation and sound quirks(it has a really cheesy theme song. Really cheesy), you'll find an amazing series.