If you are a BT Broadband customer you automatically have free access to the FON network of access points. There are other ways of joining as well, for details explore the links on that site.

If you look up Whitby in the "Find WiFi" link on that page you'll see there are oodles of FON points mapped.

Some of the cafes have access available, but I'm afraid I don't know whether they charge. There is also an internet cafe at the Coliseum.