There's not all that many railway museums in the area - the Yeovil Railway Centre springs to mind, but AFAIK it's only fairly small. There is also the new M Shed at Bristol, which I believe has a section about the railways, but it's not solely transport based.
In terms of preserved lines, the WSR is the obvious one being a couple of miles up the road. There is also the East Somerset Railway, but I'm not sure what it's like or whether it's worth the journey. The South Devon Railway at Buckfastleigh is easy to get to - less than 45 minutes down the M5 and A38. The Paignton and Dartmouth Railway is also a great day out, especially if you get the ferry across to Dartmouth itself. Getting there would probably be easiest by train from Taunton or Tiverton Parkway (depending on where exactly you're staying).
Slightly further afield, there's the Lynton and Barnstaple Railway just the other side of Exmoor - not a full day in itself but from Taunton you'll probably want to go via Lynton and Lynmouth where you can take the Cliff Railway. If you're prepared to travel a bit more, then there's the Swanage Railway in Dorset (probably just over an hour away).