Louis, you have a lot of fun time ahead of you! I envy you that you have not ever watched the show yet because it will be a real discovery for you. Personally I think The Next Generation (TNG) is the easiest to get into for most people. It certainly has the largest fanbase of all the series. The Original Series (TOS) is what I grew up watching in the 60's and it will always have a special place in my heart but I much prefer the later shows. My personal favorite is Deep Space Nine (DS9). It's quite a good show with an excellent cast, writers and stories. It is darker than the other Trek series and yet still Star Trek. Voyager is a fine show for what it is and I think it gets run down way more than it deserves. Enterprise (ENT) had some moments of brilliance but comes in last on my list of favorites. However, I am a fan of all the series and not like some fans who hate certain Trek shows and refuse to watch them. Finally in the TV shows there is also the Animated Series (TAS) and it is somewhat like a 4th season of TOS using most of the same actors for voices. It's a little cheesy by todays cartoon standards but I still think it's fun!
Like others above I would highly recommend watching all the movies in release order and with little time between viewings if possible. Each movie somewhat builds on what went before and I have found that I get the most out of them if I don't have long gaps between watching each one.
I'm glad to know that there are still people out there discovering this great show for the first time! I hope the new movie coming this next Christmas will bring in a whole new generation of Star Trek fans and fandom.