Many Gundam fans consider Zeta their, by far, favorite series. It's dark, it's depressing, many character relationships, and a lot of old favorites (like Amuro, Char, and even Katz) unite under the same banner.
A New Translation refines and gives glory to the aging series, for a new generation. As discussed in previous posts, much of it got a nice face lift, but allow me to summarize what to expect (along with my sole gripe).
The series primarily opens and follows the path of Camille, a young boy driven by his emotions. Living under the opening oppression of the Titans, Camille steals one of their prototype gundams (the MKII), and is recruited by the Anti-Earth United Government (AEUG). It is here that we begin seeing familiar faces. All characters suffer and witness many hardships through their battles and grow stronger in their commitment and abilities (mainly Camille developing his Newtype abilities).
Having seen the original episodes that correspond with the movie, I can tell you not much has been left out or changed. A few scenes are shortened, and the ones left out feel more like a deleted scene extra that many DVDs have. Also, only one major plot point was delayed until later but it still eventually occurred. My only gripe is that things do begin moving faster and faster, but not enough to get lost (until III, which I'll comment on as well).
If you've seen the original Zeta, definitely pick this up, it'll be nostalgic seeing it in modern glory.
If you've seen other gundams, but never Zeta, be hesitant. If you're still enthusiastic about seeing Zeta, try movie I, and if it interests you, pick up the rest. If this route is taken, be prepared to look up additional summaries and FAQs as the trilogy progresses (I do believe MAHQ has the original Zeta episode summaries).
If you're new to anime and wanna see a gundam after something you saw on Cartoon Network, do NOT begin with this, you'll be beyond lost. Try watching the UC gundams in their time line order: 0079 (original trilogy is very good), 08th MS Team, 0080 (my personal favorite), 0083, then Zeta, ZZ, Char's Counter Attack, and if you're still feeling excited, V.
If you're not into anime, why would you even be reading this?