Think of TOS (The Original Series) as Greek Morality plays on a stage.
TNG (The Next Generation) is the same, but with a budget and special effects - what Roddenberry imagined for TOS.
For someone new, I'd say start with TNG, it captures the essence of the original but is more watchable for someone who didn't grow up with the original Trek. Plus everything else really builds from it, sort of. Well, at least Voyager and Deep Space Nine are set in the same timeframe and kind of follow TNG canon.
Edit: the special effects on TNG were pretty much the best you could do for TV at the time. Remember, a large TV then would've been 20" or so.