From Chiang Mai you can go to Lamphun by public bus or by sawngthaew. The bus station for Lamphun bus is in Eastern side of the city. But ask the ticket seller to tell where you want to go off the bus. Wat Phra That Hariphunchai, for example, is very far from bus station. Back to Chiang Mai you can go by bus from Lamphun bus station, but there are also Chiang Mai buses waiting outside Wat Phra That Hariphunchai in Lamphun.
The cheapest way to get to Lamphun or back to Chiang Mai is by blue songtaew. From chiang mai-you get one by the Ping river by the bus station. They drive by, just wave your hand and find room in one.
From Chiang Mai, Lamphun is the last stop-so you don't have to worry about price. The ride takes around 45 mins to one hour.