Apple’s support documents should guide you, but they’re a little confusing if you’re running quite old versions of OS X. I haven’t had to migrate files via Time Machine for over a year, and it’s a little bit of a complicated situation, but not a horrible one. You can do this during system setup of a new Mac, after installing a fresh copy of OS X on a computer, or even using Migration Assistant while OS X is up and running and you’re logged in. Time Machine backups can be used to do a full system restore of a dead hard drive or computer, but also migrate just user files, applications, and some settings from one computer to another. If I was to upgrade to a newer, or even the newest OS, will the new OS be able to use the Snow Leopard era Time Machine backup to restore my things on the updated OS? Or am I doing backups more or less for nothing?
Eric Desrochers happily runs Mac OS X 10.6 Snow Leopard-and that’s a version that a lot of people remained delighted with for a long while, particularly the 10.6.8 final release.