The occasional campiness of the 1960's Star Trek series was not at all evident in Star Trek: The Next Generation, arguably the best science fiction television series to ever be aired. It's funny TNG wound up so brilliant, considering the first two seasons were a bit iffy in terms of quality. The show ended up the ultimate representation of Star Trek, with an immensely engaging crew with a great sense of camaraderie, intelligent and intriguing stories, and special effects that were excellent by television standards.
With the exception of Wil Wheaton, the cast was uniformly superb. Patrick Stewart had a lot to live up to as a successor to William Shatner's Kirk. With his magnetic presence and wonderful acting chops, he's crafted a very different individual from Kirk, and probably my favorite Trek character, period. Jonathan Frakes made for a likeable, intriguing Commander Riker, who's occasional rowdiness reminded me a bit of good old Kirk. Brent Spiner is simply terrific as the android, Data, who aspires to be more human. I can't imagine anyone else in the role, which is probably the highest praise you could give to an acting performance. Michael Dorn excelled as the tough Klingon Worf, Levar Burton was immensely likeable as chief engineer Laforge, and rounding out the cast were the series' two sexy and smart women, Marina Sirtis as Counselor Troi and Gates Mcfadden as Dr. Crusher (the latter of whom looks even more beautiful now than she did in her first season on TNG).
The series has had a barrage of standout episodes, whether it was with suspenseful ventures into the unknown (the first appearance of the Borg) to the ultimate cliffhanger in The Best of Both Worlds, The Next Generation was an exercise in masterful storytelling and vivid characterization. Since then, three shows have spawned from Trek lore: Deep Space Nine, Voyager, and Enterprise. I have only seen one episode of Enterprise (and clearly, it didn't make much of an impression on me), but I had sporadically watched DS9 and Voyager, enough to know neither of them were overall quite as compelling as TNG (the only aspect they definitely had over TNG were in the sets and visual effects). For me, that's rather odd, since I often have a preference for a TV series that follows through a central storyline, as both DS9 and Voyager have, but it simply goes to show how each stand-alone episode of TNG made an impression.