Honestly, before you shell out hundreds of dollars on DVD sets you may or may not even like, I'd go with a Netflix subscription, which will get you all five series for under ten bucks a month... plus tons of other stuff. As for disliking you just because you've only seen the NuTrek movies? So what? Everyone went through it for the first time and it's good the new movies are igniting new interest in the franchise. There will be some people here who still think anyone who didn't watch Trek in the 60's is lacking some fundamental understanding, but just ignore em.
Starting with The Original Series and the Man Trap is a perfect enough place. I'd just go with Charlie X and then on down the episode list. To me, TOS episodes are real hit and miss and admittedly to me the old school... cheeziness.. for lack of a better word, makes it hard to watch at times, but there are great acting performances.
If you want, you can start with The Next Generation which is more modern, though even it's beginning to look dated. I easily consider Picard the finest of the Star Trek captains and Patrick Stewart is fun to watch no matter what role he plays. He's not the action hero captain like Kirk though. I definitely enjoyed TNG more than TOS, but that's personal taste and it will vary with everyone. Though I wouldn't watch Deep Space 9 or Voyager until you get through at least some of TNG. And Enterprise, despite it being a prequel, is really best watched last of the five Trek series. Though Deep Space 9 is the most serialized of the five series, having a more solid plot over it's seven seasons than any of the other series. Not to mention Trek's best characterization and storytelling in my opinion.
Another alternative if you're wanting some quick TOS Trek fixes is just watching the first six movies. You can watch those without having seen any of TOS and still follow what's going on quite easily. Either way, have fun with your first voyage through the Trek world, and keep us updated!