Chiang Mai is the town’s closest airport; allow 40-45 minutes to get there and expect to pay 500 baht by taxi.
Lamphun’s tiny railway station is located a couple of kilometres north of town. You can of course hop on any Chiang Mai train heading south for Phitsanolok and Bangkok, but the railway’s also practical for short trips to Lampang, Khun Tan or Chiang Mai itself. The special express services have limited seating so need pre-booking, but for short rides you ought to be able to just turn up and get a third class seat on the slower services. For instance, the 10:06 would be the train for Khun Tan or Lampang.
Lamphun's bus station is located just south of the town centre though bear in mind many of the Bangkok to Chiang Mai express services only stop on the main highway outside of Lamphun, necessitating a motorbike taxi into town.
Conveniently there is a 24-seater bus doing the Chiang Mai run that leaves from in front of the National Museum on Inthayongyot Road. The air-con mauve and white buses take around an hour, cost 25 baht and depart every 20 minutes between 06:30 and 18:00. (Their terminus is Chang Puak bus station.)
For longer haul routes the reliable Nakhonchai Air buses departing Chiang Mai stop here on route to Udon Thani, Khon Kaen, Phitsanolok and so on. Their booking website is only partially in English (and doesn’t seem to include Lamphun anyway), so the best thing to do would be to just pop along to their office at the bus station.
Blue songthaews connect outlying districts as well as neighbouring province such as Lampang and Chiang Mai, and leave from various point around town as well as the central bus station. They’re very slow – so pointless for say Chiang Mai (20 baht if you do want to do it) -- but they are your only means of getting to Amphoe Li (Mae Ping National Park) or the scenic Mae Tha district.
You could of course hire one for a daily rate for Khun Tan, for example, which is practical if you have a few people. Rates obviously depend upon distance, time and negotiating skills, but we got one for Chiang Mai via Hang Dong for 600 baht. Try the guys hanging out around the market rather than going all the way to the bus station, or your hotel ought to be able to help.
We were unable to find any motorbikes for hire, so bring your own from Chiang Mai. The khanom cheen restaurant on Ratchawong Road does have two bicycles for rent at 50 baht per day, while Phaya Inn offers free bikes to residents.
Tuk tuks haven’t reached Lamphun yet so you'll have to make do with either old-style rickshaws (or cyclos) or motorbike taxis to get around Lamphun. A rickshaw will set you back 40 baht or so for a short ride around town, while 60 baht should take you a bit further afield to Wat Chama Thewi or Wat Phra Yeun. There is also a small electric buggy-type ‘bus’ (or overgrown golf-cart) which leaves from Wat Phra That Haripunchai and does an hour or so's tour of nine of the town’s main sites, leaving you a few minutes at each spot. It's a pretty good deal at 100 baht per person.