How can I install vanilla Android on the Kindle Fire? I just received my Amazon Kindle Fire and since I am not located within the USA I can't use most of services provided Amazon. So I think the best thing for me would be to install a vanilla (custom) Android ROM.

How can I install vanilla Android on the Kindle Fire?
The most stable option at this point is to use CyanogenMod. You can get it onto your Fire by doing the following:
Root and Custom Recovery
On Windows, you can use the Kindle Fire Utility to root the Kindle Fire and install Team Win's recovery (TWRP). It's extremely straightforward, and should even set up the Google drivers for you. The whole process is done via a command-line numeric menu.
On non-Windows OSes (of if you just prefer it) you can manually root with BurritoRoot. It's also fairly painless. You'll then need to set up adb on your OS, and install TWRP, which can be done by following this guide on RootzWiki. The gist of it is that you need to reboot into fastboot and then flash the recovery image, which should be:
user@linux ~ $ adb shell
$ su
# idme bootmode 4002
# reboot
user@linux ~ $ fastboot -i 0x1949 boot path-to-twrp.img
user@linux ~ $ fastboot oem idme bootmode 4000  &lt;-- After auto-reboot
user@linux ~ $ fastboot reboot

Or something along those lines.
Flashing CyanogenMod
This works just like any other recovery system, really. Just grab CyanogenMod  and put it onto your /sdcard partition. Reboot into TWRP by powering off the device, then pressing/holding the power button at the FIREFIREFIRE bootloader (installed with the TWRP image), and use TWRP to select and flash the CM zip.
Cyanongenmod for Kindle Fire

CM10 / Jelly Bean
Ice Cream Sandwich