The one where they save the whales is IV. I have been known to tell people "even if you don't watch anything else with the original crew, you should watch movie four", because truly it is the pinnacle of Original Trek. Leonard Nimoy is a genius. (Of course, I like to get them to watch the whole II-III-IV trilogy if I can, just because it's so incredibly tightly plotted for something that was written as a series of oneshots!)

Q is... a lot like the Doctor in some ways, actually, now you mention it. Kind of if they played up the alien side of the Doctor a lot more - Q is really the completely incomprehensible alien, who usually winds up having done something helpful and important, but (on TNG) you never have any idea why he actually turned up in the first place. That's something that never really works with the Doctor, because at least since Two took over, he's been definitely benevolent; when Rusty tried to play Ten as the unpredictable force of nature last Christmas, we just all went "WTH? The Doctor's gone wacko!"

(I actually have a theory about Q's motivations... naturally, it's just as cracked-out as the man entity himself. It's my opinion that Q is the ultimate fanboy - a fic writer who likes AUs and takes advantage of his omnipotent powers to see the real characters acting out his stories. He can be a bit of a Mary Sue when he's using self-inserts for exposition, but he does allow the cast to keep their own personalities, thus vastly improving the quality of the "fic" beyond what your average Sue would produce.)

TNG does tend to be very srs bsnss, seems as if, from the few eps I've seen yet. They like to Address Issues, and also go quite heavy on the hard science/timey-wimey side of things... but as that's what I like in my skiffy (along with a side order of goofiness which Riker, Data, and Q between them can usually be relied on to provide), I'm loving it.