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'm not sure that "Encounter at Farpoint" (the TNG pilot) is something I would recommend to any newbie. Maybe "Measure of a Man" which was possibly the first truly classic episode of TNG.
For DS9, "Duet" is far and away the best ep of Season One--and a good indicator of things to come.
VOYAGER, on the other hand, had probably the best pilot of any of the latter-day shows. A nicely-paced introduction to the series and its characters.
Meanwhile, I hesitate to recommend certain "change-of-pace" classics like "The Inner Light" or "The Visitor" or "Far Beyond the Stars" just because, as memorable and creative as they are, they're far from representative of what those series were usually like (which is part of the reason they're so good). Great eps, but not ones you'd want to use to introduce somebody to Trek . . ..