Firstly: Prayer in congregation in the mosque is obligatory for both residents and travellers alike. Please see question no. 45815.
Secondly: prayer in congregation in the mosque is only obligatory for those who are near the mosque and can hear the adhaan from the mosque without the aid of loudspeakers, because of the report narrated by Muslim from Abu Hurayrah who said: A blind man came to the Prophet (peace and blessings of Allaah be upon him) and said: “O Messenger of Allaah, I do not have any guide to bring me to the mosque,” and he asked the Messenger of Allaah (peace and blessings of Allaah be upon him) for a concession allowing him to pray at home, and he allowed him that. Then when he turned to leave, he called him back and said: “Can you hear the call to prayer?” He said, “Yes.” He said: “Then answer it.” Narrated by Muslim.
Whoever is far away from the mosque and cannot hear the call without loudspeakers is not obliged to pray in congregation in the mosque. See question no. 20655. If there is such a distance between your house and the mosque, then there is nothing wrong with you offering the obligatory prayers and taraaweeh in congregation with your wife at home, in fact this is better before Allaah than your praying alone.
The reward is greater, for both obligatory and naafil prayers, the more worshippers are present in the congregation.
It was narrated that Ubayy ibn Ka’b said: The Messenger of Allaah (peace and blessings of Allaah be upon him) said: “… a man’s prayer with another man is better than his praying alone, and his praying with two men is better than his praying with one man. The more people there are, the more beloved that is to Allaah.” Narrated by al-Nasaa’i, 843; Abu Dawood, 554. See al-Silsilah al-Saheehah, 1912.
Shaykh Ibn ‘Uthaymeen said:
The Prophet (peace and blessings of Allaah be upon him) described a man praying with another man as being better than his praying alone, which implies that it is greater in reward.
As’ilat al-Baab il-Maftooh, question no. 1464.
For more information on praying taraaweeh in congregation please see question no. 21740.