Learn Russian: Ты берёшь абонемент с собой в фитнес-клуб? - Are you taking the pass with you to the fitness club?

Собой is the Instrumental case form of себя, the reflexive pronoun meaning "oneself". In Russian, when an action reflects back onto the subject of the sentence (here, ты), you must use the reflexive pronoun. So, you are literally asking "Are you taking the pass with yourself?" Using с тобой here is grammatically incorrect because the subject and the object of the preposition are the same person.

Берёшь means "you take", but the infinitive is брать. Where does the е come from?

The verb брать (to take) has an irregular present tense stem. Instead of following standard rules, it adds an е to the root and uses ё in the endings. You conjugate it as я беру, ты берёшь, он/она берёт, etc. Because it is a very common verb, it is best to simply memorize this spelling change!

Since фитнес-клуб is a location, why doesn't it have a Prepositional case ending here?

Because the sentence is talking about direction toward a destination, not just being located somewhere. When the preposition в means "to" or "into" a destination, it requires the Accusative case. Since фитнес-клуб is a masculine inanimate noun (not a living thing), its Accusative form looks exactly the same as its dictionary form.

Could I change the word order and say Ты берёшь с собой абонемент...?