Forget about what was just said about using special software like Carbon Copy. Too complicated to set up for your need and honestly I have no idea what they want you to accomplish with them especially.
Simplest way : In your current Snow Leopard system : plug in the USB Drive, go in Time Machine and select it as your whole system backup drive. It will backup EVERYTHING on your computer. May take some time (hours). By default it won't exclude anything, but in the settings you can choose to exclude particular drives or folder you don't want to back up, but really don't bother with that.
When it's done. You can safely upgrade your machine to Mountain Lion.
If you use the standard "directly from the App Store" method, you shouldn't even lose any files in the process, since OS X will automatically keep everything and simply "upgrade" the OS to Mountain Lion. You should have everything already set up when you boot up the first time in Mountain Lion. The Time Machine Backup is simply a precaution if something goes wrong in the process.
However, some peoples prefer to do a "clean install", which mean that we completely format (erase) everything on the original drive and then install the OS with only the default things set up. Honestly, Apple don't seem to think its important, but I have to admit that the system seems a bit more responsive and it's easier to manage exactly what is installed or not, what is compatible or not with the OS. For example, if a third party App is going to be of some problem with the new OS, it's easier to find it by reinstalling everyone of them from scratch. Anyway, I still think that a "clean install" install is a desirable thing to do from time to time (like every one or two years). If you want to do that :
http://osxdaily.com/2012/07/25/how-to-clean-install-os-x-mountain-lion/
Then, with a clean install, you can still go and browse your old Time Machine backup for your files and manually copy what you still want. You may also use the "Migration assistant" software, but if you do, I suggest you use it only for user settings and files and avoid using it for App, which is the reason why a clean install is an interesting thing to do in the first place.