If you purchased a Kindle Fire at any point before Amazon refreshed the lineup in September, I've got bad news for you: none of the improvements made to the software used by the 2012 Kindle Fire and the Kindle Fire HD are going to be brought to your tablet. If you have any particular complaints about the way your older Fire works or acts, that's too bad, because it's the end of the line.
Luckily, the Fire's low price and popularity relative to other Android tablets has made it a common target for Android's bustling open-source community, which has automated most of the sometimes-messy process of rooting and flashing your tablet. The Kindle Fire Utility boils the whole rooting process down to a couple of steps, and from there it's pretty easy to find pretty-stable Jelly Bean ROMs. A CyanogenMod-based version is actively maintained, but I prefer the older Hashcode ROM, which is very similar to the interface on the Nexus 7.
If you need extra help, Liliputing has a pretty handy guide to get you through everything—bear in mind, however, the standard warnings about rooting tablets. As we'll soon see, installing different software on your Kindle Fire can greatly increase its speed and utility, but it doesn't come without caveats. If you choose to do this to your older Kindle Fire, know that you're definitely voiding what's left of your warranty, and though I've had no major issues with the tools above, you do run the risk of turning your tablet into a useless brick. Some Android features may work intermittently or not at all, and there may be software bugs that the maintainers of these ROMs never get around to fixing. There are also security concerns, since unlocking your tablet's bootloader can make it that much easier for it to be screwed up by malicious code or, in the case of theft, used to access your personal data.