Stagecoach in Hampshire operates a bus from Andover to Winchester every 3 hours. Tickets cost $7 and the journey takes 30 min. Alternatively, you can take a train from Andover to Winchester via Basingstoke in around 46 min.
From Andover, Bus Station line 85 bus to Winchester, Worthy Lane. 30 min £4. From Andover train to Basingstoke then train to Winchester. 46 min £37. From Andover taxi to Winchester. 20 min £44. From Andover drive to Winchester. 20 min £3.