Kanchanaburi is about 3 hours drive from Bangkok, along Hwy 4 (Phet Kasem) from Bangkok until it meets Hwy 323. This will take you all the way to Kanchanaburi.
You can catch a taxi to Kanchanaburi, return to Bangkok for the day for around 2,000 baht. This should include stopping at the Bridge over River Kwai and museum, Kanchanaburi township, the local dam and the cemetery for the prisoners of war. You may need to pay a bit extra to visit Erawan Falls which is about an hour out of the town centre and the Tiger Temple