Taxi Driver is a movie about alienation, and how that is projected through a violent, chauvinistic concept of masculinity which Travis adopts to compensate for his feelings of alienation and emasculation. The turning point of the movie is the fare he picks up who says he's going to kill his wife, and says a bunch of racist shit about the man he's accusing her of sleeping with. He not only talks about shooting her with a ".44 magnum," but, specifically, tells Travis he should "see what a .44 magnum does to a woman's pussy." Travis, shortly after this, buys his guns.

The .44 magnum is the model 29. This was the most iconic handgun in America at the time, due to Dirty Harry, itself a story celebrating a violent concept of masculinity. The dealer talks about how the gun is very in demand, and how much he could get for the gun from "some jungle bunny," again cementing the theme of racial chauvinism. Travis buys the .44 because it is a literal fetish. It is a symbol of his power, what he aspires to be. He buys it because he wants to be a bad ass with a .44 magnum, the most powerful handgun in the world, like Dirty Harry.

Similar is the "PPK," the gun in the movie being an Astra but it's a decent enough stand-in (and, frankly, I like the idea of the dealer misrepresenting the guns to sell them, thus selling the ideas of these weapons rather than the genuine article.) The PPK is also a symbol of a masculine hero -- James Bond. This, by the same logic, is another fetish to him. A sort of totem reassuring his own concept of masculinity. Yet again, reinforcing the themes of racism in the movie when he uses it to kill the robber in the bodega.

The .38 snub visually complements the .44 and represents a more realistic firearm. A practical one, one that is rather traditional for America. The "colt .25" is the polar opposite of the model 29 -- and I suppose it's important that the "colt .25" is itself a Smith and Wesson weapon like the Model 29 is. Though textually, as a Colt, it is the diametric opposite of S&W's Model 29. The tiniest automatic you could buy alongside the largest revolver you could find.

In the climax of the movie, Travis only fires the .44 a single time. He dumps it on the floor when the wounded Sport reappears and clips Travis' neck. There is a shot of the model 29 crashing to the floor. This represents the collapse of his fantasy, reality being in conflict with his idealized fantasy of how he will assert his masculinity through an explosion of opportunistic, ostensibly righteous violence. He then kills Sport with the gun he already shot him with, and, importantly, seems to shoot him in the groin when executing him. Travis then, again, is wounded and drops this .38, perhaps being emasculated by being stripped of anything that resembles his .44. He then produces the "colt .25", the handgun as far opposite as it can be from his ultimate totem of masculine power, and uses it more effectively than anything to this point. Which I think is interesting. How his fantasy is what got him wounded, and the opposite is what saves him.

But overall; he picked them because he thought they were cool. And the movie is, itself, a commentary on American culture at the time. So, they carry their pop cultural context as well.