"Encounter at Farpoint" (1987)
The Next Generation 1x1/1x2
First appearance in the first episode of Next Generation, Q famously puts humanity on trial, calling it a barbarous race and highlighting WWIII as evidence that its so-called progress is a sham.
"Hide and Q" (1987)
The Next Generation 1x10
What may be surprising to note in the early appearances is that Q is obsessed not with Picard but Riker, another indication that Number One was all but a fallback plan in case audiences didn't embrace the older, bald captain who was nothing like Kirk (and Riker was in contrast similar in almost every way). This is the one where Q actually drafts Riker into the Q Continuum.
"Q Who" (1989)
The Next Generation 2x16
The episode better known for the debut of the Borg is really about Q pushing Picard to admit that humanity has more to learn than it may be willing to admit (which makes this probably a Vulcan's favorite episode). He offers to become a member of the Enterprise crew, but finds plenty of opposition in that ambition, notably from Guinan, who at this point is still pretty mysterious, and apparently has a history with Q that is never really explored.
"Deja Q" (1990)
The Next Generation 3x13
The difference in the creative approaches between the early seasons and later Next Generation may best be understood by the contrast of approaches and lack of continuity between this appearance and previous ones. This is the second time in the last two visits where Q has to make a case for joining the crew of the Enterprise, only this time it's because he's human, having been punished by the Continuum for past misdeeds. Although of course that would probably be the perspective of a creative team who don't appreciate what other writers were trying to do with Q, and instead see him as the same nuisance as Picard does.
The Next Generation 4x20
Q next is firmly invested in his adversarial relationship with Picard (notice the now-complete shift away from Riker and to the more entertaining verbal jousts with the captain), meddling in his relationship with Vash and spending most of the episode in a Robin Hood fantasy that most certainly has nothing to do with his prior concerns.
"True Q" (1992)
The Next Generation 6x6
Probably the least-known appearance by Q is this one that subtly shifts his focus back to the Continuum's interest in humanity, as we meet a Q who doesn't know she's a Q, thanks to the peculiar circumstances surrounding the death of her parents. This may also be considered the basis for the Continuum crisis later featured in Voyager.
Deep Space Nine 1x7
Frequently deemed the most frivolous appearance (then again, every appearance by a Next Generation personality in the first season of DS9 was deemed frivolous by impatient fans) of Q, this is a sequel to "Qpid" and clearly an attempt to contrast Sisko with Picard. Will always be fondly remembered by me for the line, "You hit me! Picard never hit me!"
The Next Generation 6x15
Although perhaps the single greatest appearance by Q in any series, this is also the one with the least to do with the Continuum's interests. It's merely a reflection of Q's identification as a quasi-antagonist for Picard.
"All Good Things..." (1994)
The Next Generation 7x25/7x26
The trial finally concludes, ending Q's story in Next Generation on a note that has absolutely nothing to do with any of the appearances past "Encounter at Farpoint." But a nice bit of continuity for the series.
"Death Wish" (1996)
The suicide episode that puts Q on the defensive when a fellow member of the Continuum proves an even bigger maverick than Next Generation sometimes wanted Q to seem. One of the few instances of fans begrudgingly respecting Voyager.
"The Q and the Grey" (1996)
A direct sequel to "Death Wish" (all of Q's appearances in Voyager are linked), the Continuum is in turmoil and only a radical solution can heal the wounds of omnipotent existentialism. Turns out Janeway won't have to mate with Q, though, because he's got a girlfriend in the Continuum, just one indication that all those Next Generation appearances did not explain everything about him.
Q's son completes the arc begun in "Q Who" (and continued in "Deja Q") when he's forced to become a member of Janeway's crew for a week or be expelled from the Continuum. What does Q really think of humanity? What does the Continuum? In summation, and thanks to "Q2," one might say that in the end it's a relationship of begrudging respect.