These recomendations are pretty good, but there's one series in particular I see left out. Gundam SEED. I've seen quite a bit of anime, and I'd say that Gundam SEED is one of the greatest series ever.
Oh, yes, we definitely need a Gundam title on this list. And I think you picked the best one to put on this list. What makes "Gundam Seed" so interesting is that it combines several elements from other Gundam Series (not that I've seen them all - I get much of my anime from Cartoon Network). So if you can only see one Gundam, I'd say watch "Gundam Seed".
Let's see if I can make a summary here - In the year 70 of the cosmic Era, tensions between the newly created coordinators (humans enhanced by manipulating their genes) and naturals were mounting and eventually erupted into a full scale war. Since the nuclear attacks did devastating damage to both sides, N-jammers were developed to neutralize nuclear weapons. The weapons of choice in the war then became mobile suits - human shaped battle machines about 18 meters in size armed with various types of weapons.
The story starts in year 71 of the Cosmic Era as Kira Yamato and several other students at the space colony of Heliopolis are near a secret weapons development area when the coordinator forces (called ZAFT) attack and try to forcibly seize the new weapons. Seeing his friends in danger, Kira (the only coordinator among these students) pilots one of these mobile suits - the strike - and fends off the attack from ZAFT. The strike and the battleship Archangel are saved although several other mobile suits fall into the hands of ZAFT. Heliopolis is damaged to the point of destruction and must be evacuated. Kira and the other students exposed to the new military weapons become the new crewmen of the Archangel (as several others were killed in the battle) and hurry to link up with the naturals' fleet, pursued by ZAFT.
Although Kira's only goal was to protect his friends, he is now involved in the war between ZAFT and the naturals. And the fact that he is a coordinator makes it strange that he is fighting on the side of the naturals.
New groups/characters are introduced as the story progresses. And groups/characters change - old friends become enemies and old enemies become friends. But that's how the story begins. For more info., here's the official website:
http://www.gundamofficial.com/worlds/ce/index.html