This 一番に is an adverb that means "first(ly)". に is a particle that makes the previous word adverbial. For example 永遠 means "eternity" and 永遠に means "eternally". 元気 means "energy/spirit" and 元気に means "energetically". Likewise, 一番 means "number one" or "the first/top", and 一番に is its adverbial version.
Those te-forms are just connecting several actions, namely 起きる, 集まる, and (太極拳を)する.
He arrived first. / He was the first to arrive.
The first thing my family do in the morning is gather in the courtyard and do tai chi together.