Why does he take her to see a porn movie?

In addition to what others have said it's as if his self loathing has gone to such an extent that his subconscious has to destroy the only potentially positive thing in his life.

It's part of his sickness. He wants what he thinks he can't have and alienates himself from what he could have.

Taking Betsy to that theater is him proving himself right about his special alone-ness and credit Betsy's rejection to the fickle feminine nature he uses to excuse himself from normal interaction with women.

"Who can't use the Force now?! I can still use the Force!" - Yarael Poof

To be fair, back in the 70's it was hip for couples to go to pornographic theaters. Many of them were quite fancy too, especially the Pussycat Theater.

The keyword here is "couples". If Travis would've waited until they went on a few more dates, Betsy probably would've seen it as a wild adventure.

I think Travis was just so dazed and confused, that he actually thought it would be suitable to take her to see a porno movie. He lived a really strange and lonely life.

Him taking Betsy to the porn movie...it seems to me more of a matter of stupidity than social inadequacy.
I mean, a lonely person is not THAT stupid. A man, no matter how lonely he is, knows (more or less) what type of movies women like (or PROBABLY would like).
Besides, I never thought of Bickle as crazy, he seems (painfully) lucid most of the time. I think his violence comes from his personal frustrations and his will to help Iris, not because of madness.
The scene with Betsy at the porno is really weird, because MOST of the time Travis doesn't seem to be a fool. I always though that in a sense, he wanted the date to be a complete failure, maybe because subconsciously he knew that relationship wouldn't last too long.

Yes, he truly thought she’d be into that. He wasn’t trying to offend her (he liked her legitimately). He just didn’t know that everyone in the world finds porn offensive. To him, it was no different from a romantic comedy or a Disney musical. The scene helped demonstrate how disconnected he was from the common man.

The scene also highlighted his loneliness. If he didn’t already know that he shouldn’t take a girl to a porn movie on a date, then he hasn’t gone on many (if any) dates in his life beforehand. Otherwise, he would have known better. Porn also has strong intrinsic associations with loneliness because it helps men who cannot get real sex with a real woman.

Travis spent a lot of time watching porn alone and equated it with human intimacy. To Travis, pornography was something special to him, and he tried opening up to Betsy by taking her to see it, which he was hoping she’d understand.

For some reason, he didn't find that as inappropriate. He was really screwed up. But yeah, I would feel the way his date feels where if someone took me to a porno compared to other movies that are out, he's just doing this to get me in the mood to sleep with me.