How is the Tarawih prayer performed at home during the month of Ramadan? Is it prayed in congregation or individually? And should the recitation of the Quran be audible or silent?

You may pray Tarawih in congregation with your family, which is preferable. It is also permissible to pray it individually. There is no disagreement regarding the permissibility of congregational prayer for Tarawih and Witr in Ramadan. Voluntary prayers may be performed in congregation or individually, and the Prophet (peace be upon him) did both. As for vocalization therein, the Imam vocalizes, the one praying alone vocalizes moderately, and the follower does not vocalize at all.