Ok first of all, this question is too naive. I don`t mean to be disrespectful in any way but Can I explain something to you.
First of All Apache Tomcat is the servlet container that is used in the official
Reference Implementation for the Java Servlet. In other words you are going to be developing in JSP Java Server Pages technology.
Before you begin to develop I would suggest you do some research to see if .jsp is the correct approach to building a dynamic web site. As for Mysql, yes it`s free most web hosting companies offer mysql as their default open source database.
BTW.. I host Apache Tomcat and Mysql, it`s the same as a regular Linux account.
Hope this solves your mystery
Web Developer / Owner / Consultant
When starting your own business the need to "know-how" is greater than money!