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...we've already got one of these. But fine.
Get started with the Mobile Suit Gundam movies. If you like them, and want more, you could move on to the full series, or watch the Zeta movies. I'd advise against watching SEED or 00 until you have a grounded understanding of the series... even though they aren't actual sequels, you'll appreciate some of the content more having seen the older series. Also, people who start with 00 or SEED tend to become retarded.
If you want something less 'real' and more 'awesome', try GaoGaiGar, Getter Robo, or Gurren Lagann.
If you want some teenage psychosexual angst and oedipal complexes mixed in with your mech (and a healthy dose of psuedo-religious symbolism) watch Neon Genesis Evangelion.
If you want to watch guys in powered armor on motorcycles beat up other guys in foam monster suits, try Kamen Rider. The series do not need to be watched in order, some are grimdark, some are downright silly.
You also can't go wrong with Macross, The Big O, Escaflowne, Bubblegum Crisis, or any of the other dozens of shows that pop up when you google 'mecha'.