This manual contains a series of hands on tutorials that expose the user to the features in VisIt. The first three tutorials form a good basis for using VisIt, including the basics of using the Graphical User Interface (GUI), performing data analysis and using Python to script and automate tasks in VisIt. After that are a series of tutorials that cover advanced topics in detail.
Here are links to the datasets used in the VisIt Basics, Data Analysis and Scripting tutorials.
Here are links to the data used for Aneurysm (Blood Flow) simulation tutorial, which is also used in one of the examples in the Scripting tutorial.
Here are links to data used for the Potential Flow tutorial.