> I just downloaded The J2SE Development Kit 5.0 (JDK TM 5.0) from the Java website.
> Is this the right one for programming?
It allows you to compile your Java source files.
> If this is the right one for programming how do I actually open up Java and start programming?
Bottom left corner of your desktop.
Start -> Run -> [ notepad ]
start typing
It is the right one for programming but the JDK does not contain a full blown IDE in the regular download (you can download a version including netBeans as well.
Apart from your advanced editing needs, the JDK contains what you need to develop and run Java applications. You will mainly need javac (to compile .java files into .class files) and java (to run those .class files).
I recommend you to read at least this:
http://java.sun.com/j2se/1.5.0/docs/tooldocs/index.html
While you are finding out how it all works, like trying HelloWorld and slightly more complex samples, yes it is generally recommended not to use an IDE yet so that you get familiar with issues like classpath.
Once you get how the basic toolset works, you can and should use a more advanced IDE like NetBeans, Eclipse, ...