Websites: Start by testing every hyperlink, any user account and comment functionality, test functionality of any forms or scripts. Move on to more advanced tasks depending on the functionality, use and design of the website.
Test on multiple browsers on PC in different resolutions, as well as options for Chrome and others on Android, iOS Safari, the Kindle Fire browser and the browser on the PS3.
Games: Full test of all inputs via keyboard/mouse, Xbox 360 pad, generic controller and PS3 controller. Test level design (make sure it flows properly without bugs as well as making sure the level is as airtight as possible to stop player from falling off) and any logical issues with the actual design itself (would mostly be useful from a critique position but maybe I'll spot something you didn't).
Apps: Test Android applications in a similar fashion using an APK on Kindle Fire and Nexus 4.