So by now you have your environment variables set, you are ready to start Tomcat and look at the examples that come with it. To do this, run the startup.sh or startup.bat file in the TOMCAT_HOME/bin directory. On Linux you may have to add executable flags to the .sh files.
Once you've started Tocmat, you can now go to http://localhost:8080 and see the default install. Spend some time looking at the example JSPs and servlets.