I am new to Ubuntu and I would like to uninstall many of the programs I don't use that came with version 11.10 (e.g. games, Tomboy notes, Scan, etc). Though, I am scared of the thought that by doing so - when I upgrade to a new Ubuntu version - my computer might get broken. Is there reason to fear? Or is the Ubuntu upgrade procedure bullet proof?
No.
Uninstalling software that you don't use or want will not burst your PC into flames, cause an earthquake or destroy the planet, so don't worry as long as you don't Uninstall needed software in which case you have to know what exactly you're uninstalling.