Greg's list is perfect (though not really a fan of Devil or Arena; just a bit too cheesy for my taste). All the Trek series are dated now and have their problems but they're easy to get into for anyone who has more than a passing interest in character and story. Yes, even VOY and ENT.
regarding series 'tasters', why not watch pilots of each series? All bearable. Though the DS9 one is a bit weird and the TNG one meh.