The most popular hike in the region of Kazbegi is the visit to the Gergeti Trinity Church. Starting in Gudauri you would take a taxi ride to Stepandsminda. Its about a 45min drive. Tell the driver to drop you off a bit up the road towards the church.
You can tell the taxi to drive all the way up to the church but I recommend the little hike up it should take you about one hour. Wander Lush mad a nice Trinity Church Hike: A Comprehensive Guide