Well that very much depends on you, as each series is different.
The original - Classic stories, holds up quite well. It's action on the edge of the frontier, with one man in charge, leading the way.
The Next Generation - Follows up the original some 80 years into the future. Technology can do anything now, including create people (holodecks). They are explorers, still, but it's more of a committee, though the Captain is still in charge, they have board meetings in nearly every episode, to discuss things and think them through.
Deep Space 9 - Another series set on the edge of the frontier, but they don't really go anywhere, as it's on a space station. There is a wide variety among the cast, as it's the largest cast of the series. There is more or less an overall arc to this series, and it's best to watch from beginning to end. It's not necessary to see the other series to enjoy this.
Voyager - Two ships get flung to the other side of the galaxy, one gets destroyed, so the two crews must work together to get home. They are under the command of a female captain (the only female lead in any of the series). This captain's morals are ambiguous at best, and change more frequently than her hairstyles.
Enterprise - This is the most recent series, and takes place 100 years before the original series. The idea is that there was an Enterprise before Kirk's Enterprise, and things are more primitive. Taken on its own, without the Trek connection, it's an ok series. When viewed as part of the Trek series, continuity with the other series is nearly non-existent.
My suggestion to you is to go to Netflix and rent the first dvd from each of the series. Watch the pilots for each (with the exception of the original series, each pilot is two hours long, the original was one hour). See which you like best, and continue from there.