‘ And so it was that on the next day the people rose early and built there an altar, and offered burnt offerings and peace offerings.'
The ‘building' of an altar refers to refurbishing it since its last use and preparing it for the offering of sacrifices. The ‘rising early' demonstrates that they were in earnest. Then again they offered burnt offerings and peace offerings. The burnt offerings were offered wholly to God. The peace offerings they could partake of themselves. The one represented total dedication, the other their fellowship with God through the shedding of blood.
Bethel was one of the places where God had revealed His name in the past, and He had done so again in Judges 20:28. This last was presumably the grounds on which they had rebuilt the altar. Thus they were proceeding in their view in accordance with Exodus 20:24.
Then they began the serious business of dealing with those of their own who had failed to observe the covenant.