Note that the suggestion to use IE to look at the Temporary Internet Files folder wasn't intended to suggest that you need to use it for browsing, if you're happier with another browser. It's just the simplest way to get to the TempInt folder in most Windows versions. Other browsers should be using the same folder for cookies and temporary web files, since Windows defines that as the place those things should be.
It is worth noting once again that IE is an intimate and non-removable part of Windows, and even if you don't use it you should keep IE up to date with security patches. IE and Windows Explorer are in reality the same program, just wearing different masks for appearance purposes. Windows Explorer gets many, if not most, of its updates as IE patches, and Win Explorer is the main way of getting access to all of the files on your computer.