For getting started just download the Neo4j Server and start it. Then go to http://localhost:7474 for the integrated web-admin UI which allows you to enter data visually and browser/visualize and query it.
Please have a look at the Neo4j Koans by Jim Webber and Ian Robinson which are material that are used in real-world tutorials. Otherwise also have a look on http://video.neo4j.org for some screencasts and presentations and the collection of introduction links at the neo4j delicious site.