Welcome to the forums!
General steps are root, flash a custom recovery (often this is included if not required for root access), make a Nandroid backup (a copy of the OS in case something goes wrong), download and install the ROM.
Now, you may be limited in options depending on your device. You can check out XDA for most ROM options, or you can look at the websites of MIUI, or Cyanogenmod. Granted Cyanogen is not MIUI, but another good ROM.