The nearest train station to Coniston is at Windermere, which is 15 miles away. This is only served by a local branch line. Mainline trains call at Oxenholme where you can connect for Windermere. There is some sort of bus service from Windermere to Coniston but I'm not sure how regularly. An alternative would be to get the train to Ulverston, on the Furness line, from where there is a bus service to Coniston.
National Express buses call at Ambleside, 8 miles from Coniston, again with a local bus service connection.
A bus service also operates from Ulverston to Coniston and Kendal to Coniston.