The reason I love Star Trek is nothing to do with any particular character. It is more to do with the concept, the title shots are what really do it for me, especially in TNG and Voyager.
The vast expanse of space yet in fact though so vast there IS life out there. There are so many thousands of lifeforms, from the Q continuum to Vulcans to ourselves having developed as a species beyond what we were. It hints at truths of the past being that we were visited by aliens, that we finally attain a degree of conscience that warranted our stellar capabilities, not just a warlike species invading everywhere but one that could be part of a Universe and find its place in the wider existence. Yet despite mysteries being unveiled, so many more existed. To travel in a ship like the Enterprise and to see those things, to be part of new discoveries and learning about these new things and seeing a galaxy being born or ended but from right beside it almost. THAT is what makes Star Trek great because it so very well manages to capture the very concept it started as and improved on it and made it believable