All the previous suggestions are good (though Unicorn, while phenomenal, should probably be viewed with a prior knowledge to the UC setting). A good starting point would be one of the good "gateway series", particularly Gundam Wing or Gundam SEED. Both are standalone but do their best to encapsulate the "spirit" of Gundam. I have my issues with both, especially SEED, but they're a decent jump-on point that can then lead to experiencing the other (better?) stuff.
As a sidenote, the best Gundam series I have ever seen is 0080: War in the Pocket. while it takes place during the original series, next to no prior knowledge is required. It's only 6 20-minute episodes, too! Add to that the fact that it's probably the most emotionally impacting, down-to-earth, and well-written entry in the franchise, and you have a short war drama that I would recommend to anybody.
But let's say you want to experience the franchise from the first series, Mobile Suit Gundam. You have a few options there.
The original series, which I actually think has stood the test of time fairly well, is readily available online, either dubbed on sites like Crunchyroll or fansubbed. There's also a movie compilation version, which cuts the whole thing down into 3 feature-length films.
There are pros and cons to both; the original series invented the genre of Real Robot, in that it was the first robot show that tried to tell a serious sci-fi story, but as most pioneer series do, it retains a few of the elements of the genre it grew out from (in this case, the fun-lovin' Super Robot genre). That plus old serial animation make it occasionally hard to take completely seriously. The movies try to fix this by using the bigger budget to produce better (and more consistent) quality animation, and by trimming out the silly stuff. However, trimming up the series to fit a movie length means that sometimes events don't flow as well, and you get less time to know the characters. I prefer the series, personally, as I find the movies themselves tend to drag a bit.
The manga that this thread is the subject of is actually shaping up to be the best version of the original story available; not only is the art phenomenal, retaining the old style of the show (it's done by the same artist) and being incredibly detailed and expressive, but it's a good balance between the series and the movie, keeping things as serious as they need to be but still taking the time to delve into characters. In fact, it adds quite a bit of backstory that is in neither (and is thus of dubious canon). The only problem is, you can only get the content of the first 6 or so volumes in English; the series has been canceled and is no longer being released by Viz. It's being scanslated, too, albeit slowly.