Is there a good, fast browser (faster than Firefox) that I can use on an old computer?
You didn't mention, but I'm assuming this old computer is running Windows - if it's a Mac, not much in this answer is going to be much use.
If you're willing to sacrifice even more of the visual fidelity of websites, you could try a really lightweight browser such as Lynx (text only browser) or Dillo (graphical, but not very stable on windows). Further options after that would involve installing some variant of Linux on the system, which will give you access to a wider range of lightweight browsers, but this would be a fairly drastic step.
Other people asked questions on similar topics, check out the answers they received:
Other people asked questions on various topics, and are still waiting for answer. Would be great if you can take a sec and answer them