By plane
Lampang Airport LPT is currently served by one daily turboprop ATR-72 flight by Bangkok Airways (http://www.bangkokair.com) from/to Bangkok Suvarnabhumi airport, with a stopover in Sukhothai. The flight takes less than 1 hour from/to Sukhothai from 1600 baht and around 3 hours, including stopover time, from/to Bangkok 2400 baht and up.
The cheaper option is to take a low-cost flight Air Asia, One-Two-GO, Nok Air to Chiang Mai, then connect to Lampang by bus see below.
By train
The Northern railway line from Bangkok to Chiang Mai passes through Lampang. As such, there are several trains in either direction every day. The journey from Bangkok takes 10-12 hours; 2nd class air-con sleeper ticket costs 734-804 baht, air-con seat car is 574 baht both can be booked online (http://www.thairailwayticket.com/Default.aspx?language=1). 2nd class fan sleeper is 150-200 baht cheaper than aircon, but can be booked only at the station.
From Chiang Mai, the trip takes around 2 hours. Express fan car tickets are 23 baht. Air-con car tickets are just over 100 baht and all tickets can be bought at the station only.
By bus
There are many buses going from Mo Chit bus terminal in Bangkok to Lampang. Many of those are continuing on to Chiang Mai. First-class aircon bus ticket costs around 400-450 baht, the trip takes 8-9 hours as such, overnight buses are recommended.
From Chiang Mai Arcade bus station, buses are very frequent, and it takes just around 1-1,5 hours to get to Lampang. The price is 40-80 baht, depending on bus class. as at Nov 2010, first class aircon bus costs B90
Finally, many buses going from Chiang Mai to various other destination in Central and North-Eastern Thailand have a stop in Lampang. As such, it is usually possible to board a direct service to get where you need to.