Can they be done and how is it done?

Looking to change from a Tourist Visa to a Non-B

Ideally just inside Thailand, but I am open to suggestions on what would be the easiest way to do it.

Found a new job that will supply the paperwork for me.

Or if I have to leave the country, where is the cheapest and easiest place to do this?

To apply for a change of visa status to a non immigrant visa (class B ) you will need all the documents required to do an extension of stay. In Bangkok there is one additional requirement for a certified education certificate.
You you should check with the office where you apply to find out what their requirements are. Where you apply will depend upon which region of the country you live. See: New Locations For Obtaining Non Immigrant Visas Based Upon Working And Etc
You will need a work permit application approval letter to do the change or a apply for a visa from an embassy or consulate.
In northern part of the country Laos would be the best choice and Penang for the south.