Two buses per day run direct from Bangkok's Morchit (northern) bus terminal to Sangkhlaburi, but at seven to eight hours (the last four of which are quite a climb) the ride can be a bit daunting. Most travellers opt rather to make the connection in Kanchanaburi, from where a range of buses and minibuses run to Sangkhlaburi throughout the day.
Sangklhaburi-bound buses depart from the main bus station in central Kanchanaburi every hour from 07:30 to 16:30. Air-con buses cost 180 baht and take four hours while the non air-con red local buses cost 100 baht and take at least five hours.
Minibuses also depart hourly from 07:30 to 16:30 from the Kanchanaburi bus station, costing 146 baht and taking three to four hours. Since the minibuses can get rather crowded, they're not a good idea if you have a lot of luggage.
In Sangkhlaburi, air-con buses and minibuses depart from the east end of the main market road while non air-con red local buses and songthaews depart from the bus station, which is located down a sidestreet across Si Suwan Khiri Road from Thetsaban 2, not far from the post office or market. Note that all buses make a stop in Thong Pha Phum if you're trying to get there. The schedule at time of writing is as follows:
Bangkok (Morchit): Departs at 07:30, 10:30, 14:00 and 15:00, costs 274 baht for 1st class or 213 baht for 2nd class and takes seven to eight hours. You can also catch these buses to Kanchanaburi.
Kanchanaburi (air-con bus): Departs at 08:45, 10:45 and 14:30, costs 180 baht.
Kanchanaburi (red local bus): Departs at 06:45, 08:15, 09:45 and 13:15, costs 100 baht.
Kanchanaburi (minibus): Departs at 06:00, 06:30, 07:30, 11:30, 13:00, 15:30 and 16:00, costs 146 baht.
Three Pagodas Pass: Songthaews depart every 40 minutes from 06:00 to 18:40, cost 30 baht and take 35 minutes.
Sangkhlaburi's accommodation area to the south, Mon village and town centre can make for a lot of walking as they're all fairly well spread apart. If you like walking, you'll be fine, but the area is perhaps best explored by bicycle, which can be rented at J.Family Homestay or P Guesthouse for 70 baht or Oh Dee Hostel for 100 baht per day.
Motorbikes can also be rented at P Guesthouse and J.Family Homestay for 200 to 300 baht per day.
There are also a handful of motorbike taxis, usually found hanging out near the market in the town centre. And blue songthaews can be chartered at the bus station for daytrips into the hinterland. Prices could range anywhere from a few hundred baht to over 1,000 depending on how long you want to go for, how far you want to go, and how willing you are to haggle.