Let me bid farewell to my family: This statement is similar to the prophet Elisha's request to Elijah (1 Kings 19:19–20). Jesus asked for an even greater commitment from his followers.
puts his hand to the plow and then looks back: Farmers in the ancient world guided a light plow, a tool used to prepare soil for planting, with one hand and the oxen (work animals used for farming) with the other. Looking away would cause the plow to turn away.
For a believer, looking back meant placing earthly concerns ahead of God.