You do know that Mozilla (the group that puts out Firefox) has a technical forum for exactly this type of question?
Firefox Support • mozillaZine Forums
You can find out the version of Firefox you are using by going to Help>About Firefox.
Since the 3.x series of Firefox they have had "Private Browsing" mode, which doesn't store image files, cookies, a url history, or any of that on the hard drive. You can get to that by pressing Control-Shift-P, or Tools>Enable Private Browsing. (Internet explorer version 8.0 also has a private browsing mode, as does Google Chrome -- though theirs is very slow).
If you want to access the cache you can do that by typing "about:cache" in the url window, and that will list the items viewable in cache.
While its true that a small amount of information is still cached to the hard drive (for windows memory management), this generally isn't in a form that is useful for anyone to determine your browsing history. If you are really concerned about that kind of thing, you can invest in a little bit more memory for your computer, some ramdisk software, and move the windows cache, and even your browser cache to the ramdisk (and then its gone the second you turn the computer off).
Another big security hole is stored login information/passwords. If you use that feature, those sites are going to be retrievable that you have visited if the right software is applied. (IE stores these in the Windows Registry -- just in case you were curious. I'm not sure where Firefox does, but they are retrievable if you don't set a master password for your account from within Firefox itself.)
You might also want to be aware that if you have a gmail account, or Google toolbar that you may have enabled storage of your search terms unintentionally, and anyone who knows your gmail password can access that quite easily.
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To some degree I have the same issues you do. My SO made an offhanded comment to me one day that could only have been raised by a good deal of snooping around, and it got me thinking that I needed to know the type of person I was dealing with. So I designed a test to measure her propensity to snoop by leaving a window open, logged in, and turning logging on. She obsessively went through every piece of email on my machine she could get access to, and even used it to retrieve a password for a site I visited occasionally. I respected the effort.
I wasn't put off by this. In an odd way I felt a lot better since she wanted to pry to know more about me. Since I knew about it, and knew her propensity to pry, and some of her anxiety issues I did something very left handed: I put all of the accounts for my email, im, social networking, and many boring technical sites with passwords, and login names on a card in my wallet in the section where the money, and receipts go; so I knew she would find it. I'm sure she has them by now, and can browse them on her computer at her leisure. Security through openness.
Anything I want to keep private, I simply have sent to another set of email addresses I created after, and I came up with a completely new set of passwords. Anything I truly want private that needs to be stored, I encrypt into a hidden volume, and if uber paranoid about it, put it on portable storage marked as porn or something else I know she'll want to look at. I use private browsing occasionally when it suits my needs. I figure she'll get bored with the checking up on phase sooner or later, and that it makes her feel better to think she has that in her bag of tricks for later.
For the opposite problem: How to track activity.
If everyone uses their own accounts this works out great:
Microsoft gave a fairly useful tool for parents in the form of parental controls. If you have windows Vista, or Windows 7, and have administrator access, you have a fairly good set of tracking tools. You can simply turn on logging under parental controls. It came with the OS. Its also fairly decent at keeping your kids off of pr0n sites...It didn't stop my 10 year old from bypassing it by logging onto grandma's computer to visit pornhub though.
If they do their browsing somewhere else, it wont help you.
Its not completely transparent. Any account with administrator access can find out its turned on for their account, but the reports are only sent to the account that enabled it by default.
It will keep a log of what software is installed on the computer, what software is accessed, every time an administrator override is required, and does a fair job of tracking site viewing with a very simple report that gives you the most viewed, most visited, last sites visited. Its also useful for figuring out what the heck people did to break things when fixing technical problems. (I really hate the "I didn't install anything new" line when I CAN go check...) The handy thing is, you can be lazy with it, and don't even have to log out of your own account to use it. Just check the report, and done.
Other methods:
You can dig through the history yourself, but I prefer to look at cookies, and stored passwords (IE stores login names, and passwords the in the windows registry in several different places for those that are truly curious where it puts them). The browser cache is also useful for a run through, but its better for finding things like porn than finding the sites they came from at times.
If I really want to be nosy, I could just mirror the router, and I can watch traffic in real time. (You need a little better than a consumer router for this most of the time...) My network, my rules. Believe it or not, most of the traffic goes out in plain text, even from major IM services. I block certain types of traffic at the router level to keep the kids off of certain sites & services that are problematic, but I can also choose to log this traffic at the router level as well. If you control the network, other than encrypted traffic routed though a proxy server -- you can have a good idea what is going on in the network if you know what to look for in a good router, and how to set it up. Not worth the time involved, but if you want to geek out its easy enough to do. This is how they do most monitoring in work environments, and its very effective for network traffic.
While there is software out there that can go deeper than this, its use straddles the wiretap laws. I don't employ conversation logging for IM's, or keystroke recording, as it runs into wiretap laws. Not even for the kids. I simply block the protocols I don't want at the router level for their machines, and have a short interval for the automatic lockout for the other computers in the house to keep them off of our accounts.
I have very little interest in what she does with the PCs, other than to mine. My square headed Medusa girlfriend is mine alone...my precious...
If they have physical access to the machine, any security effort you do is largely meaningless if they *really* want to know. That being said, if its something like your own laptop that isn't shared, a boot password is a good idea. (You can set this in the BIOS, and its a good idea to set a BIOS password at the same time, and lock out booting off of DVD/CD/Flashdrive/Network). This cuts down greatly on the ability of anyone to install software without your permission, or noticing. Just don't forget those passwords. You have to take the laptop apart to reset them!