Why does Gene take the bike ride to the beach when he feels he has logical reasons to remain at school?

Gene goes with Finny for several reasons:

1. First, he is enamoured with Finny.  Despite Gene's jealousy toward Finny and his belief that Finny might be trying to sabotage his academics, Gene still wants to be like Finny.  He might have even thought that if he spent more time with Finny, perhaps Finny's confidence, skills, and likeability would wear off on him.

2. Gene is not unlike any other teenager--he is far away from home, has immersed himself in his classes, and longs for an escape.  Very few people have not played hooky at some point in life, and Finny presents an opportunity for Gene to escape the pressure of proving himself at Devon and to the other boys.  Plus, if Gene goes with Finny by himself, that raises his standing with other students at the school.

3.  Finally, I think that Gene wants to prove to himself that he can loosen up a little and be the "bad" boy for once.  This is connected to #1 in the sense that he longs to be as carefree as Finny is.  He might have convinced himself that if he can blow off his test and go have fun for a day that he is on his way to changing who he is.