The girl obviously wants to have the baby. The man obviously does not want her to have the baby, although he knows very well that she wants to have it. She knows that he doesn't want a baby because it would change their lifestyle. He likes roaming around and looking at new things and trying new drinks. If she had the baby they would have to settle down. Living in Europe was very cheap, and Spain was one of the cheapest places. But if they had a baby they would need a fixed place of residence and he would undoubtedly have to get a job. Getting a job in Europe might be very difficult. Since he is an American, the natural thing for him to do would be to go back to America and take the girl with him. This would almost certainly mean that they would get married, because in those days she wouldn't want to have a baby without being married. So the American is not only pressuring her to have an abortion because he doesn't want a baby, but he probably doesn't want to get married either--assuming they aren't already married. When the girl says, "That's all we do, isn't it--look at things and try new drinks?" she is suggesting that they have done enough of that, that it is time to move on to a different stage of life involving adult responsibilities. She is implying that she has done that with him, gone everywhere he wanted to go, but she is getting tired of their nomadic way of life. Being pregnant will change a woman's whole outlook on life, making her want a home and security. The man can understand this, but he doesn't want to understand it. He thinks he is being "reasonable," but he is just as emotional as she is. Practically everything that is going on between them is beneath the service--more in what they don't say than in what they do say. Their talking about hills that look like white elephants is a way of not talking about what is really crucial.
Actually, you quoted incorrectly:
"I wanted to try this new drink. That's all we do, isn't it--look at things and try new drinks?"
This has a totally different meaning when you understand the context. The drink they order is "Anis del Toro".
What the "new thing" the girl is looking at is the mountains. "I said the mountains looked like white elephants."