Why would Travis Bickle go on a date to an adult cinema?

There are two things here. One is social ineptitude, where Bickle isn't aware this is socially offensive.

There is another aspect of this, given the timeframe. The film Deep Throat in 1972 had been a big hit and shown in normal US cinemas, and porn had (almost) become acceptable:

...alongside a newly made documentary about the movie called Inside Deep Throat which examines its making and the effect of its taking porn into the mainstream.

From the Independent - 'Deep Throat', the biggest porn film ever, is cleared for release

So it wasn't so unusual, especially in New York, and time has made it more shocking.

Travis does so because of

• Social ineptitude
• He is clueless about how inappropriate it is
• But he knows it's inappropriate and likes the idea of angelic Betsy in that dirty theater

Quotes from the screenplay by Paul Schrader.

About Travis (and Betsy)

Travis lives in his own world and has a unique lifestyle separating him from others. He is isolated, especially from people like Betsy. When observing Betsy from his Taxi, he is described

He is like a lone wolf watching the warm campfires of civilization from a distance.

He was in the army unti May 1971 (he may have arrived in NY not too long ago), he is not educated (at the beginning: "Some. Here and there.") and his growing obsession with cleaning the city doesn't help either.

He is unsusual in many ways, e. g. he can't sleep at night, his willingness to "work anywhere, anytime" regardless of danger, just watching TV or later bringing himself back to shape to assassinate Palantine. That's not always bad, though, since it makes Betsy become interested in him in the first place (what a contrast he is to her immature colleague). See scenes 26 and 28, for example, where it's described what draws Betsy to him.

His social ineptitude is also shown in scene 11, when he awkwardly tries to hit on the concession girl of the porn theater

He is obviously trying to be friendly - no easy task for him. God knows he needs a friend.

But overall, he doesn't keep up to date, shows no interest in politics (read scene 27 when Betsy interviews him because he wants to volunteer and he is clueless about Palantine, or scenes 38-39 with Palantine in his cab: "I don't know, sir. I don't follow political issues much.") or culture (not just the porn theater, but also right before, when he gives Betsy the Kris Kristofferson record: "I don't follow music much. I'd like to though. (second thought) Honest."). And he doesn't really care about it, his interest is only due to his interest in Betsy, saying he wants but never does.

In the porn theater

Scene 41:

When Betsy realizes that he really wants to take her to a porn movie, she is horrified, but he tells her that many couples go there, so he believes it's socially acceptable.

Travis seems confused. He is so much part of his own world, he fails to comprehend another's world. Compared to the movies he sees, this is respectable. But then there's also something that Travis could not even acknowledge, much less admit: That he really wants to get this pure white girl into that dark porno theatre.

Note the sarcasm in the following:

Travis escorts Betsy to an empty center row. Travis was right. Couples do go to films like this. There are at least six or seven other MEN with their bewigged "DATES".

And when she leaves, the following dialogue

                     BETSY
        These are not the kind of movies I
        go to.

                     TRAVIS
        Well, I don't follow movies too
        much...

                     BETSY
        You mean these are the only kind of
        movies you go to?

The TICKET GIRL watches expressionlessly from the booth.

                     TRAVIS
        This is sort of high class...

                     BETSY
        I mean porno movies.

                     TRAVIS
               (hesitant)
        Well... mostly...

Note how he admits to not following movies too much. He only knows porn movies.

Travis Bickle was an antisocial, poverty distraught, shell shocked, drug addict. As he begins to degenerate into oblivion, he philosophizes that:

Nothing can... touch... her...

And puts absolute faith into this woman, as a symbol for the good in the world.

As a beacon of the good in society, Travis believes that this love he feels for Betsy will save them both from a tormented existence.

I'll tell you why. I think you're a lonely person. I drive by this place a lot and I see you here. I see all these people around you, and I see all these phones, and all this stuff on your desk... That means nothing.

This act goes beyond the realms of audacity as Travis's character reveals a reverence for humanity and ethics.

Betsy confides in the authenticity of Travis’s good will.

...There was an impulse that we were both following. So that gave me the right to come in and talk to you. Otherwise, I never would have felt that I had the right to talk to you or say anything to you. I never would have had the courage to talk to you.

Travis did not blink during this statement.

He revels in the joy of the lingering fascination with Betsy.

...Betsy... Betsy... I forgot to ask her last name...

On the second date when Travis takes Betsy to the pornographic theater, he assures her that:

A lot of couples go to see these types of movies.

These experiences had become such a normal part of Travis’s life, and that he thought Betsy’s moral imperative was just understanding, thought to bring the person he idolized to show her the evils of the world in which he fought.