It is, then, an emphatically recommended sunnah. Performing it in the mosques and in congregation is better; if, however, a person prays it on his own or at home there is no harm in that, though praying it in the mosque behind the imam, together with the congregation, is superior, more complete, and more excellent.
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The Tarāwīḥ Prayer and the Tahajjud Prayer

Question: What is the ruling concerning the Tarāwīḥ prayer and the Tahajjud prayer? When exactly is the time for Tahajjud, and how many rakʿahs does it comprise? If someone has already prayed Witr at the end of Tarāwīḥ, is it still permissible for him to perform Tahajjud? Must Tarāwīḥ be joined immediately to the ʿIshāʾ prayer, or may the congregation agree to delay it, disperse, and then reconvene later to pray Tarāwīḥ? Is such a delay permissible or not?

Answer: As for the Tarāwīḥ prayer, it is an emphatically recommended sunnah, and it is to be performed straight after the ʿIshāʾ prayer along with its regular sunnah units. This is the practice of the Muslims. Delaying it, as the questioner suggests, to a later time and then coming back to the mosque to pray Tarāwīḥ runs contrary to that established practice. The jurists state that it is to be offered right after ʿIshāʾ and its regular sunnah; so if people do postpone it, we would not declare the act unlawful, but it would indeed be at odds with what has long been done. It is a prayer of the early part of the night—this is the settled practice.

As for Tahajjud, it too is a sunnah and carries immense merit; it is the night vigil (qiyām al-layl). Standing in prayer after having slept—especially during the last third of the night, or during the third that follows the halfway point of the night—occurs in the depths of the night and brings tremendous virtue and a great reward.