First off:

Do I have to have a web server (excuse my knowledge, but I'm afraid I don't even know what a web server is)?

There's something in terms of "the basics®" you need to learn. By definition you need access to a web server to serve your pages from. Secondly,

Also, what would I use to web program in Perl?

You are probably getting way ahead of yourself. Try renting shared space on a web server somewhere. Hosting plans are as little as US$1 a month these days. Start off by learning HTML and serving up static pages. If you try to learn Perl web programming and just the task of creating HTML pages you are going to get lost.

Having said all that and once you are comfortable with concepts of HTML start checking out the Tutorials here at the Monastery. There is a whole section on web programming that if you try to learn it all at once your head will a'splode.

My advise to you is to start off very simple, follow some of the examples in the CGI man pages and don't move on until you understand them thoroughly.

Make sure that whatever hosting plan you get yourself into, or whatever server you end up posting your work to is configured to allow CGI programs to run on them. Some don't for security reasons. Some don't at the "economical" hosting plan levels because they want to charge you more for the privilege. So choose wisely.