The number 51 bus service, operated by Hatch Green, runs between various points in Stoke St Gregory and Taunton every couple of hours or so, Monday to Saturday. Their timetable may be found here. (Or a PDF version here.)
The number 29, operated by Buses of Somerset runs between Taunton and Wells, stopping in East Lyng. Their timetable is here.
The nearest stations are Taunton (10 miles) and Bridgwater (12·5 miles), for up trains to Bristol and London Paddington, or down to Devon and Cornwall.
Crewkerne (21 miles) and Yeovil (28 miles) are further away, with routes to Salisbury, Bournemouth and London Waterloo.
There are currently no taxi services within the village, but plenty not too far away.
There is no formal car scheme in the village, but plenty of people willing to give a lift to those in need. If you are unable to drive and need help getting to a health centre appointment for instance, get in touch with Graham Gleed (01823 490407).