I wouldn't recommend watching only the Zeta Gundam: A New Translation movies by themselves. They're not that great, compared to how well the MSG movie trilogy conveyed the message of the original series. I'd recommending watching the series first to get an idea of what the story was originally intended to be, then watching the movies to see what was changed, and to appreciate that you're rewatching Zeta in three short movies with newly animated action after the entire TV series, rather than the other way around which would seem longer.
Old animation is kept with the new animation. Sometimes, it can be a bit jarring to see a battle in the very nice, newly made animation be "interrupted" with a cut to a scene that was drawn in the '80s. They either should have made it with entirely new animation, or leave less of the old animation. I don't mind the old animation at all, it just doesn't flow well when they cut back and forth from old animation to new animation. It gets better with each movie, though. I'd say the first movie's got more old animation than new, movie 2 is about half-half, and movie 3 is pleasantly mainly new animation, and the old animation doesn't stick out as much.
Some important story elements are cut out for the sake of time. There's no Dakar story arc, arguably the most important talking scene in Zeta Gundam because Char basically outlines the AEUG's objectives, tells the Federation assembly what's wrong with the world, and reveals how the Titans really were. Another important story arc that was cut, deals with a character close to Kamille, and is a parallel to an event in the original Mobile Suit Gundam, but I won't mention specifics for spoilers.
I'd say that the movies change quite a lot, although the main theme and message remains the same. The ending is quite a surprise if one has seen the TV series first, although it is largely the same. I'd say watching both is the best way to see the story. I figure that they left the '80s animation as a nod to the older people who saw the show on TV and would probably go again to see it in theatres for the nostalgia.