Learn Russian: Потом они пошли по старому мосту.

It follows the preposition по, which requires the Dative case when describing motion "along" a route or surface. Here, it tells us they are walking along the surface of the bridge.

How do we form the endings for старому мосту?

мост is a masculine noun ending in a consonant, so it adds -у in the Dative case to become мосту. The adjective старый (old) must match its noun, so it takes the masculine Dative ending -ому to become старому.

Why use the verb пошли instead of the imperfective шли?

пошли is a perfective verb. It is used here to show a completed action in a sequence: first something else happened, and "Then they walked..." If the sentence used the imperfective шли, it would mean "Then they were walking..." and focus on the process instead of the completed step.

I remember the word мост having unusual stress. How is мосту pronounced here?

The stress shifts to the final syllable in the Dative case, so it is pronounced мостУ.