This might be a really trivial question, but I've been writing Java programs at my school and I just found out that I can create websites with Java as well.
Read the tutorial on Java Web applications.
Basically Web applications are a part of the Java EE standard. A lot of people only use the Web (servlets) part with additional frameworks thrown in, most notably Spring but also Struts, Seam and others.
All you need is an IDE like IntelliJ, Eclipse or Netbeans, the JDK, the Java EE download and a servlet container like Tomcat (or a full-blown application server like Glassfish or JBoss).
Here is a Tomcat tutorial.
You are asking a few different questions...
The simplest way to start making websites with Java is to use JSP. JSP stands for Java Server Pages, and it allows you to embed HTML in Java code files for dynamic page creation. In order to compile and serve JSPs, you will need a Servlet Container, which is basically a web server that runs Java classes. The most popular basic Servlet Container is called Tomcat, and it's provided free by The Apache Software Foundation. Follow the tutorial that cletus provided here.
Once you have Tomcat up and running, and have a basic understanding of how to deploy JSPs, you'll probably want to start creating your own JSPs. I always like IBM developerWorks tutorials. They have a JSP tutorial here that looks alright (though a bit dated).
You'll find out that there is a lot more to Java web development than JSPs, but these tutorials will get you headed in the right direction.
This is a pretty subjective question. PHP and Java are just tools, and in the hands of a bad programmer, any tool is useless. PHP and Java both have their strengths and weaknesses, and the discussion of them is probably outside of the scope of this post. I'd say that if you already know Java, stick with Java.
MySQL is better suited for web applications, as it is designed to handle many concurrent users. You should know though that Java can use MySQL just as easily as PHP can, through JDBC, Java's database connectivity framework.