It's a matter of personal preference too. GameOn pretty much mentioned the chronological watch order.
TOS for example is extremely dated. Now if you don't mind that, there are some good episodes there and it's the foundation on which the rest of it's built on. There's no serialization so you really could literally watch these in any order without being confused.
Honestly... unless you really just loved the heck out of TOS, I'd skip the Animated Series. That's just me though, so try a couple to see if it groves with you.
The TOS movies are probably their best hour, and for the most part they all tell one continuous story. Other than this though, TOS isn't very serialized at all.
TNG introduces a new crew in a more modern time. It's less dated, but still showing it's age. There's more continuity than TOS but the show still is basically a random adventure of the week, though their are various mini-arcs over the years and they do tie back often enough to what's been done before. The TNG movies were average overall at best.
DS9 gets off to a slow start to find itself being set on a space station as opposed to a ship, but it has some of the best characterization and storytelling of the franchise. The tone of the show is more dark, the characters are less perfect and everything is more... real... for lack of a better word.
Voyager basically takes Star Trek and mixes it with lost in space. They're trapped on the wrong end of the galaxy trying to get home for the whole series, so this basically limits them exclusively to an episodic format with an alien/problem of the week as a norm.
Enterprise also starts off weak but gets better. It's a prequel, the premise being Earth's first long range spaceship and they're all alone pretty much exploring space for the first time.
The reboot movies... flashy effects and great action sequences but plot telling and characterization is weak.
Your mileage may vary, but it gives you something to start off of. If TOS is boring you after an episode or two, fast forward to the movies, or to TNG. Sure all the shows tie together can subtly tie together especially TNG/DS9/VOY being all set in the same time period, but they stand alone just fine too. I've had the best luck converting people to Trekkies by showing them DS9 first.
Good luck, and of course welcome to the board.