How to find out whether or not a page is cached? I have some serious problems with the performance of my Drupal website and I think that the problem may caused by the fact, that the pages are not cached. How can I find out whether or not a page was cached and if not, how can I find out why the page is not cached?

I am using varnish for my page.
Whether a page is cached

You can enable logging and use varnishlog, varnishstat, etc. See Debugging Varnish for a bit more information. There's also a lot of information on Pantheon's Working with Varnish, I'd highly recommend reading that.

Why it is not cached

That's a much bigger, harder question to answer. It's entirely down to your server/application logic. Your server's VCL files, in conjunction with response headers sent from your Drupal application, will determine whether or not a page is cached. 

There may be other environmental factors affecting things, but again those would be specific to your server and your sysadmin (or whoever installed/set up the server) should be able to tell you about them.

There's a pretty decent tutorial available that walks through setting up Varnish with Drupal, I'd recommend reading through that and comparing your own set up to that known-to-be-working one.