I have some problems with fog, and I want to back it up to a newly fresh installation.
I would like to know if there is a how-to or the procedure to do so.
Depending on the type or problems you are having will probably determine what you want to backup and what you want to discard.
You can export most of the host, images defs, group, and other information from the database. You can backup your /images directory to another location, and you can copy your .fogsettings file so you know what you chose during the install.