Read the docs. Most answers are already documented.
Know the Servlet and JSP specification. Many answers are not documented by the Tomcat team since they are generic and already defined by the specs. Know the specs, they let you ensure your webapps are portable across different servlet containers.
Please do some research. Use a search engine. It is frustrating to see questions which can be answered with a simple Google search (or your favorite search engine).
Browse or search our mailing lists.

For your convenience, we have tried to break the FAQ out into multiple sections consisting of the following topics:

Bugs - You think you have a bug or there is a difference in behavior with another servlet container.
Class Not Found - What to do with Class Not Found errors.
Connectors - You want to connect tomcat to Apache, IIS, or have questions about tomcat-standalone.
Database - Getting tomcat to talk to a database.
Deployment - Questions related to web application deployment.
How To - Miscellaneous common how to's.
FDA Validation - Questions related to running Tomcat in an FDA validated environment.
Linux / Unix - Common questions for Linux / Unix related platforms.
Other Operating Systems - Information about scripts and support for other operating systems.
Logging - Common questions related to logging in Tomcat.
Memory - Common memory related questions.
Meta - About Tomcat and this FAQ.
Miscellaneous - Miscellaneous questions that weren't categorized.
Monitoring & Performance - Monitoring and Performance questions.
Other Resources - A lot of links to tomcat related documentation and experiences.
Security - Common security issues.
Version - About the different tomcat versions.
Tomcat User - More information about the tomcat-user list.
Windows - Common Windows questions.