W00t! Welcome. I was pretty taken from the start but there's no doubt that Seasons 3 and 4 are the best, and there is quite a left turn in Season 3.
definitely gets things off with a bang. A lot of things are established at once - Vulcans have been on Earth for a century. And they've kept us back. Now we're going out there. And we might not be as well-prepared as we should be. But we might also be immature enough to shrug off assistance and instead see it as suppression (but we do have some precedent for that).
The alien doctor's name is Phlox. The other crew members are the pilot, Travis Mayweather and the Armory Officer, Malcolm Reed. Plus of course there are incidental crew members but these are the big players.
In Season 1, you'll also run into a Science crewman, Elizabeth Cutler. Other crew members are mainly one-offs.