Learn German: Wir sind auf dem Berg gewandert.

In German, the Perfekt (past) tense uses a form of sein (to be) instead of haben (to have) for verbs that show physical movement or a change of state. Since hiking involves moving, wandern (to hike) takes sein. So, structurally, you are saying "We are hiked."

If hiking is movement, why do we use the dative case auf dem Berg instead of the accusative auf den Berg?

Even though you are moving, the whole hike takes place within one area: on the mountain. You aren't traveling to a new destination. Two-way prepositions like auf take the dative case (like dem Berg) when describing where an action happens, rather than the accusative for where to someone is going.

How is the past participle gewandert formed?

Wandern (to hike) is a regular verb. To make its past participle, you take the stem wander, add the prefix ge- at the beginning, and put a -t at the end. This gives you gewandert.

Why is gewandert all the way at the end of the sentence?