Bus
There are a few bus stops in Lyme Regis with the X51/53 Jurassic Coaster bus service running between Axminster train station and Dorchester & Weymouth. More details at https://www.firstbus.co.uk/wessex-dorset-south-somerset.
You can also download a pdf of the timetable here https://www.firstbus.co.uk/uploads/maps/jurassic%20coaster%20leaflet%20may21%20WEB.pdf
Train
The nearest mainline train service is from Axminster Train Station operated by South Western railway. It runs between London Waterloo and Exeter. As of July 2021, trains arrive from London at 3 minutes past every hour on their way to Exeter. Trains to London depart at 6 mins past the hour. More details at https://www.southwesternrailway.com/