Ok this is what made me a trekkie.
Watch The Next Generation from start to finish.
Then watch Deep space nine start to finish.
These two shows shares a lot of cast, and their stories are shared some. Also they are the absolutely best shows. Voyager never lived up to its name and Enterprise never really went anywhere. TOS (the 60s show) is fine and all but I'm sorry it didn't age well. You shoudl absolutely watch movie 1-6 though as they are like a small series in themselves
Anyway if you follow these instructions you are a trekkie for life.
I went from totally non caring about Trek to a very big fan