What is the ruling on someone who intended to travel and prayed the 'Asr prayer at home with his son before the time for prayer in the mosques, and then traveled?

The prayer is valid if performed after its time has begun. Performing it before the congregants of the mosque does not affect its validity. However, the preponderant opinion is that congregational prayer is obligatory. Therefore, you both committed a sin by failing to attend it, and repentance is due, unless there was a valid excuse for absence.