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.
I know I choose two first year episodes out of my three DS9 suggestions but generally aside from TOS I would avoid the first season of the spin offs. It was like the production staff and actors were intimidated by the history of Trek and its survival after decades of syndication with the rapid Trekkers thus went too far in protecting Gene's legacy and not far enough in producing entertainment.
In any event I would recommend episodes that were fun, not episodes because they established the canon.
The Year of Hell