It's not possible to reach Amphawa by train. However, Mae Khlong station in Samut Songkhram, from where Amphawa is easily reachable, can be reached via the local Mahachai line, with trains departing from Wongwian Yai rail station in the Thonburi area of Bangkok. You'll need to catch a train first to Samut Sakhon (Mahachai), then take a three-baht ferry across the nearby Tha Jeen river and catch another train to Samut Songkhram from the station immediately on the western bank of the river. In this case you would be in the train as it cruises through the legendary Mae Khlong market.
Trains to Samut Sakhon depart for Mahachai every hour from Wongwian Yai station near the BTS Skytrain station of the same name in Bangkok. In Mahachai you'll have to walk west to the ferry pier and then find Ban Laem Station on the other side of the river in Tha Chaloem. From there, a few trains depart for Samut Songkhram from 06:00 to 12:00. The whole trip takes a few hours, though most travellers spend an hour or two at Mahachai seafood market along the way. You could also take a train to Mahachai and then grab a minibus to Samut Songkhram.
Regular buses do not go to Amphawa, which has no bus station, but minibuses from Bangkok's Sai Tai Mai (Southern) Terminal run from 06:00 to 18:00 for 70 baht per person and drop off near the floating market. Regular buses and minibuses also depart from Sai Tai Mai for Mae Khlong market in the provincial capital of Samut Songkhram, where you can transfer to a songthaew (15 baht) or tuk tuk (100 baht) to Amphawa.
In Amphawa, minibuses depart for Bangkok from a clearly marked roadside stand on Phrachaset Road just east of the town centre. These run from 05:15 to 18:00 (20:00 on weekends) and also cost 80 baht.
Minibuses to Bangkok can also be caught from the bus station near Mae Khlong market (at the corner of Rattan Witthi Road and Ratpasit Road) in Samut Songkhram. Other buses and minibuses heading onwards to regional destinations like Petchaburi, Hua Hin and Ratchaburi can also be caught here.
If all of the other options sound too daunting, a taxi can take you all the way to Amphawa from the centre of Bangkok for around 1,500 baht, depending on your bartering skills.
Amphawa is a small town that can be easily covered on foot. Most hotels and homestays also offer guests bicycles free of charge. There is currently no place in Amphawa that rents motorbikes.
For getting around the general area, royal blue songthaews run regularly along the main roads, but be sure to tell them where you're headed as destinations are written only in Thai and some songthaews go north in the direction of Bang Khonthi while others run in loops from Amphawa to Samut Songkhram to Tha Kha and back. The cost to Samut Songkhram is 15 baht per person.
A handful of tuk tuks and motorbike taxis can also be found near the minibus pick-up point on Prachaset Road.
Samut Songkhram province is well connected by waterways, so most outlying sights and can also be reached by boat. The price for a private two- to three-hour boat trip is 400 baht, but one-off trips like Amphawa to Bang Noi and back are negotiable. You can also take a 3 baht ferry across the Mae Khlong river from the main pier near the centre of town, from where a few large temples -- but little else -- are within walking distance. If exploring by bicycle, you could bring your bike onto the ferry and head immediately north up to Wat Kai Bang Kung.