The first movie retells the story up until just before the final battle with Lord Genome, the Spiral King. Most of what happens will be pretty familiar to you, but the ending chunk is almost entirely new footage (while still tying into essentially the same story). Basically, the first Gurren Lagann movie is an excuse to yell at the screen with your friends or random strangers-- with a few surprises to make the already over-the-top show even BIGGER. Buy it and show it at a geek party. I can almost guarantee you'll get laid. (It just might be by your best pal who's a mech freak instead of the hot Kamina fangirl.)
Gainax did a really fantastic job piecing together the story in a much shorter package. In some cases they actually manage to make light of the fact that they can't go into the in-depth explanations or introductions to various aspects of the plot, such as the...unique navigational tool they wind up using to navigate their way to enemy headquarters. It's really good, very enjoyable stuff that adds to but doesn't muck up the whole Gurren Lagann canon, either.
The only people who should NOT see these movies (yet) are those who want to watch the TV show but haven't yet. The movies are so much about taking the TV series and doing it bigger that I think watching them in the reverse order would make the TV series seem less crazy.
The most awe-inspiring thing about Gurren Lagann isn't how creative Gainax is at creating epic battles-- although that is pretty amazing --but how much the series has resonated with people, possibly even as much as Evangelion did 13 years prior to TTGL's release. The contrast between the two couldn't have been more different, though: a major theme of Evangelion is our personal demons and failures preventing us from moving forward with our lives.
By contrast, in a time of such cynicism and pessimism, Gurren Lagann is the ultimate statement that determination and belief in oneself is most, if not all, of what we need to break through the walls we perceive to be in front of us. It's a message I'm thrilled to see connect with so many people.