On windows XP, I would open Control Panel and then open Add Remove Programs. The process is just about the same with most Windows systems. The best source of reference is Microsoft. The link below gives simple instructions on how to do this with Windows 7.
http://windows.microsoft.com/en-gb/windows/uninstall-change-program#uninstall-change-program=windows-7When uninstalling an antivirus program, make sure you are not connected to the internet. After you have removed Avast, install your new Antivirus program before connecting to the internet again.
You may need to remove some remaining items from the Avast folder in program files. There could also be other files left over in other locations such as Application Data Folder so you may wish to search your hard drive for folders that may relate to Avast. Not catching everything should not really matter though. Once your new antivirus program is up and running, and you are satisfied, you might want to do a more thorough job and use the famous CCleaner program, by Piriform, to remove entries from the Registry. Always create a backup of the registry first - CCleaner gives you the option to do so.
If the uninstallation fails for some reason, then you will need to disable Avast at startup. Here's a link to show you how to disable programs at startup.
http://www.howtogeek.com/74523/how-to-disable-startup-programs-in-windows/Disable startup programs or processes containing the words Alwil or Avast. Then reboot and try everything above once more.
Don't go online without an antivirus (of some kind) to protect your system. One more thing: you might want to purge all items in the Avast virus chest before doing all of the above. Just delete them. I'm not sure if it makes a difference, but it's best to not take any chances.