Personally, I would say star with TOS since it was the first series and it introduces us to the universe. IMHO you probably won't get quite as much entertainment out of Enterprise if you aren't already familiar with the universe and the concepts behind it. If you decide TOS doesn't do anything for you I would say start with The Next Generation, since it kind of reintroduces everything.
My personal favorite series would have to be DS9, it had amazing characters, great plots, and IMHO it probably had the best group of people behind the secenes.
Don't know if you are interested, but alot of the people who worked behind the scenes on Trek have gone on to other great shows.
Ronald D Moore (TNG writer/DS9 writer/executive producer):
started as a miniseries in 2003, series ran from 2004-2008 in it's last season(executive producer/writer/creator) IMHO one of the best shows on TV right now, won a Peabody award during it's second season.
(Exectuive producer, writer, later showrunner) ran from 2003-'05 never got to finish it's storyline I only watched a few episodes because I didn't want to get into an unfinished storyline, but what I saw seemed pretty good.
Michael Piller (DS9/TNG executive producer, writer):
The Dead Zone
(creator, writer, possibly showrunner) 2002-07 I watched this pretty regularly for the last couple years and I enjoyed it quite a bit. Sadly they just announced it won't be coming back for any more seasons. This show starred Nicole De Boer from the final season of DS9.
(creator, don't know what else)2005-present. One of only two non Sci-fi shows on this list, and one of two I've never seen.
Ira Steven Behr (writer TNG/DS9 writer/executive producer and "he deserves the most credit for Deep Space Nine's Dominion War arc")
Twilight Zone (2002-2003 series)
(executive producer, possibly more) Second show I've never seen.
(executive producer, writer) 2004-'07 Great show, often shown as a companion to DZ, also just ended.
René Echevarria (TNG writer, DS9 writer/executive producer)
The 4400 (cocreator, writer, possibly show runner)
Naren Shankar (TNG Science consultant/story edito, DS9 writer/science consultant)
(writer/executive producer) ran 1999-2003, returned in late '04 for a miniseries, and will return as a series of webisodes later this year Amazing show, IMHO one of the best sci-fi shows ever. Starred Stargate SG-1's Ben Browder and Claudia Black.
'00-present (head writer) All I think that needs to be said about this is the fact that it is one of the most popular shows on TV, for good reason IMO.