I'm trying to learn Java Servlets on my own but I'm having some very basic problems.

The code produces a compiler error, "the import javax.servlet cannot be resolved".

I'm using version 3.2.2. Why am I having this problem?

Nope by default the Java Plugin for Eclipse only includes Core Java, Servlets are not a part of that.
A short cut to resolve that error would be, in case you have tomcat / some other java enabled server installed, just check in their "lib" folder if you find any jar called "servlet.jar" or "j2ee.jar" or something similar in name and add it to your project's build path in Eclipse, this should solve the compile error.

However you will not be able to test your servlets directly via Eclipse in this case

I'm using Tomcat 5.0.

There are multiple lib folders and none of the jar files match exactly. Which folder should I look into? Even if I choose the right folder how exactly should I add it the built path? I attached the search result of "lib".

What is the other method of installing the servlet package?

if you are using the latest version of tomcat 6.0+ there is a "lib" folder in the ROOT directory of tomcat, you should find the servlet.jar file there.
In tomcat 5.0 I assume it should be in $CATALINA_HOME/share/lib The second way is to install the J2EE Standard Tools plugin for eclipse.

I found the jar file with the packages. Problem solved.