I would begin with Google Maps API. Its free and easy to work with. It uses Javascript, which can be learned via a number of good resources. I would recommend Codecademy if you are new to JS.

There are many free data sources out there which can be incorporated into Google Maps via a number of ways (KML, database, GeoRSS, etc.). Most states and many universities have free data stores which can be incorporated into your map.

One of my favorite sites for Google Maps examples is Google Maps Mania. There are great map examples which showcase what is possible in this API.