In John Boyne's The Boy in the Striped Pajamas, we are not told very much about the children's hobbies or how they spend their time. The word hobby is certainly never used; however, we can certainly deduce a few of their hobbies. One thing that is obvious from the start is that Gretel's hobby is doll collecting. However, Bruno's hobbies are described a bit more subtly as a part of his indirect characterization and don't come up until later in the book.
In Chapter 10, after Bruno decides to explore the woods and meets Shmuel behind the fence for the first time, Bruno asks Shmuel, "Do you like exploring?" (p. 62) Bruno even declares that the wants to be an explorer when he grows up and that he reads about explorers. We can therefore call exploring and reading about explorers two of his hobbies. On a rainy day in Chapter 12, he even starts reading, protesting to Gretel that reading is not boring; "it's an adventure" and much more interesting than Gretel's doll collection (p. 81). Therefore, again, we can assume that one of Bruno's hobbies is reading. In fact, later, when Kotler yanks a book from Bruno's hand in Chapter 15, we learn that Bruno is reading Treasure Island; therefore, we know that he enjoys reading children's adventure stories and not just history books about explorers.
Therefore, three hobbies we can name for Bruno are exploring, reading about explorers, and reading children's adventure stories.
In The Boy in the Striped Pajamas, Bruno is devastated when he learns that the family is moving away from his beloved house in Berlin. He has to leave his friends behind and is not convinced that the "great adventure" (ch 1) to which his mother alludes will be any such thing. The thought of not seeing his "three best friends for life" is daunting and all the plans the boys have made for the holidays will never take place. Bruno contemplates the banister which he loves to slide down and thinks about the game he plays, trying to land on his feet at the bottom of the banister or "it was five points against you."
On arriving at their new house, Bruno can hardly believe how different it is and is shocked that it is unlikely that he will even be able to make "friends [n]or trouble" (ch 2) in such a "desolate" place. The house is uninteresting with "no nooks and crannies" for him to explore and he remarks how he hadn't even finished exploring his old house after all these years and yet this one promises him nothing. He thinks about the "aromas" from the market in Berlin where he liked to stand and enjoy his surroundings and reflect on "sweetness and life"; this place is just "empty and cold."
From the descriptions so far, the reader recognizes that Bruno's hobbies (as Tamara points out above) are not really specific but are more about his preoccupation with his surroundings and what exists there. Bruno's apparent hobbies are summed up when he expresses his dismay that Mother is not allowed to insist that they return to Berlin. In chapter 2, he ponders "three best friends for life, sliding down banisters, trying to stand on his tiptoes to see right across Berlin" and contrasts it with a "cold nasty house with three whispering maids and a waiter" where it seems to him there is no possibility of "ever (being) cheerful again."
Bruno spends his time in a dream world where he is "the king... and she [Gretel, his sister] is his lowly subject" (ch 3). He even tries to tell his father about his unhappiness and points out that he cannot pass the time and amuse himself without his friends, the vegetable stalls, cafes and Saturday afternoons being "pushed from pillar to post." His discussion with his father achieves nothing for Bruno and he has to be satisfied with his curiosity about the people he has seen who all wear striped pajamas. Bruno knows he needs to find something to do at this house so he doesn't "slowly go mad" (ch 7). He looks for some rope and a tire so he can busy himself in a tree. Unfortunately, Bruno falls and hurts himself and, when Pavel comes to help him, Bruno tells Pavel that he wants to be an explorer and laments the lack of exploring opportunities in the new house. In chapter 8, Bruno tells his tutor how he likes books about "knights and adventures and exploring." He considers that it must be time to start exploring because, of all the things he used to do in the old house, he can at least explore by himself. He even dresses himself as he thinks an explorer might do and decides to explore his surroundings, especially as the house doesn't have anywhere interesting.
It is Bruno's hobby of exploring that will ultimately lead to his death with his new and only friend Shmuel.