Can I set my browser or download a special application to log/download everything that is going through network while browsing a site?

When I open a website in Firefox, it downloads the files to temporary folders, but I need everything, and by everything I mean full Ajax request logging, image downloading...etc Every file that is downloaded while browsing sites (and not temporary, I need them later, so It should log and categorize it somehow).

Otherwise: I want to automatically save everything you can see in the browser's network tab (after pressing F12) and not just the response, also need the information about what it sent and what it got back.