But I still have no idea where to start. Am I supposed to install a new site with the installer and copy some files? Is there a simple step-by-step tutorial for beginners somewhere?
Yes, that's pretty much the method. Install the new version of Plone someplace else, then copy the blobstorage and Data.fs from the old install, start the new Plone, and click on the upgrade button.
The details will come up if you have add-ons in the old Plone that you want in the new Plone. Then you have to figure out if all the add-ons have versions compatible with new Plone; if not, you have more work to do.