The first Gundam series aired in 1979-80 and had 43 episodes. The show inspired a lot of future anime and manga creators, and there are a lot of references to it in shows like Genshiken and Sayonara Zetsubou Sensei. For something as superficially mundane as a show about giant robots in space whose purpose is to sell plastic models and model kits, the cultural impact it had is immense. There is a full-size model of a Gundam in Odaiba. It used to be an RX-78, as seen in the previous link, but it was changed to a Unicorn recently.
Gundam's influence reaches outside Japan, inspiring Guillermo del Toro to make Pacific Rim. This is the cutest damn video of all time, when during PacRim promos, a Japanese morning show took him touristing and to see the Odaiba Gundam, which he didn't know they were doing. The look on his face when he sees it! That is a man having a religious experience. (Obligatory PacRim side note: I love that movie, and the sequel doesn't exist.)
So, what's Gundam about, other than giant robots in space? Each individual series has its own plot, naturally, but there are a few essential elements to make something a Gundam show, other than the robots. There is a conflict between usually a large federated space navy and a smaller faction. Sometimes the space navy is the good guys; sometimes it's the bad guys. But neither side is ever all-good or all-bad. There are corrupt bigwigs in the space navy, and there are peaceniks in the rebel factions. The lead character is an ace pilot; sometimes there's a group of ace pilot leads. The lead antagonist is an ace pilot. They pick each other as their rival/nemesis/whatever. At some point, one of the leads gets stuck behind enemy lines and this changes his outlook on the war. Fighting continues; eventually someone wins.
There are a handful of conventions that originate from 0079. The (or A) main antagonist is a blond who wears a mask and pilots a red mobile suit. This is because of Char Aznable, the Red Comet in 0079. Some Char-clones don't strictly fit the mold, but that's fine; a Char is a Char. The hero Gundam is primary colors (red, yellow, blue, white). There is a post-human type of person who has super skills with piloting and sometimes psychic powers.
Gundam and Macross are about the same age as franchises, but there are way more Gundam shows than Macross, and I haven't seen all of them. (There's a LOT! And some are really bad, and some are extra bleak.) The easiest way to break it up is to divide them into UC timeline and non-UC timeline. I'll start with UC. The UC timeline is the largest, with over 30 TV series, movies, and OVAs spanning the years 0068-0260.
Humanity has built colonies in space, mostly at La Grange points. They call them "sides." Side 3 declared independence from the Earth Federation, called itself the Principality of Zeon, and launched a war, which killed half of humanity. Eight months into a bloody stalemate, the EF Space Force (EFSF) is at Side 5 picking up their new battleship, White Base, with Zeon ships hot on their tail. Amuro Ray is a high school student whose dad is a class-A jerk and an engineer on the Gundam project, and he gets sucked in to piloting the ship and drafted into the EFSF. The ace pilot on the Zeon side is Char Aznable, who has plans of his own. This is how Mobile Suit Gundam (aka 0079) starts.
The general through plot is that Zeon Zum Deikun wanted to make/keep peace with the Feddies, but he was assassinated. The Zabi family takes charge of Side 7 and uses Zeon's name to promote their cause. Because of the pressure of living in space, some humans have evolved into Newtypes (yes, what the long-running Japanese anime magazine is named for). The Zeons want to harness the power of Newtypes to make space free from the Earth-bound government. The Feddies think Newtypes are dangerous and want to eliminate them.
The One Year War is fought to a truce in the year 0080, but politics and all that means that wars happen again and again, in 0087, 0088, 0093, and 0096. The One Year War provides a setting for a lot of side stories, most of which I haven't seen. And I will confess that I primarily know Z, ZZ, Char's Counterattack, F91, and Victory from playing the Gundam Musou games. I hear that F91 is terrible, and Victory is depressing, but not as bad as War in the Pocket (which I have also never seen, because I hear everyone dies.)
0079 exists as the 49-episode TV series and 3 compilation movies. The compilation movies get you the gist of the story, and if you want to start with a UC series, you could do worse than the movies. These are all available on Blu-Ray in the US, and the TV series is streaming on Funimation.com as of June 2021.
My favorites of the UC series are the 08th MS Team and 0083: Stardust Memory. Of the two, 08th MS Team is better as far as plotting and pacing go, but Anavel Gato of 0083 is my favorite villain.
The 08th MS Team (1996-99) is a wonderful entry point. It's a 12-episode OVA series (there's a 13th episode that was a 10-year-anniversary special, but it's not good; it's from the POV of the most annoying character) about a half-dozen Federation soldiers and their fight to defend their part of Earth from Zeon. Shiro Amada and his team are sent to reinforce a base. The local Zeon commander has a mobile armor (a really big robot, which isn't particularly mobile, because it's that huge), which is piloted by someone Shiro knows. Once this is discovered, Shiro is arrested for treason (which is the plot of the mid-series recap movie Miller's Report.)
It's moderately gritty, but not, like, Full Metal Jacket levels. The OP animation gives you a good idea of it, though it's a little cheery. This show also serves as the basis for my favorite AMV of all time, which the creator to the best of my knowledge has never uploaded, so there are only low-res uploads on YT. SPOILERS FOR THE WHOLE THING AHOY. Tenth Man Down (It also introduced me to Nightwish, so double win.)
This is available in the US on Blu-Ray and DVD via your favorite retailer.
0083: Stardust Memory (1991-92) is a side-story which is never mentioned in the rest of UC canon (for reasons explained at the end of the OVAs.) Zeon ace pilot Anavel Gato steals the prototype Gundam GP-02A, and Feddie pilot Kou Uraki is assigned to get it back from him using the GP-01. The engineer from Anaheim Electronics, Nina Purpleton, had a relationship with Gato in the past, and that causes Trouble, of course. Gato's fleet has a plan for the GP-02A, which is equipped with a nuclear warhead.
As I recall it, the pacing is a bit off, but I like it anyway. The OP and ED songs are extremely 1991. Shoji Kawamori (of Macross) did the mecha designs.
This is also available on Blu-Ray in the US via your favorite retailer. (The DVDs are 4x as expensive, so I suspect it's out of print.)