I want to have users on my website input some short info in a few fields on my website. Is there a free program that makes it easy to set this up? Do I need to have an sql database available from my host server? What's the easiest and cheapest way to save form data from visitors?

It depends on how long you want to store it, and how secure you need it to be. Sessions and cookies can be used to store data temporarily. A database would be needed for more permanent storage.