First things first, make sure you can install the app on a device that it should run on. After doing that, try and explore each bit of its functionality, ensuring that it does everything that is expected of it. This is generally the shortest and easiest part of testing as the base functionality normally works at release. The more enjoyable bit comes next.
In other words, can I break it? Here, you’ll want to try any abnormal inputs and see how it handles that. You’ll want to try everything you can to get the app to throw error messages. Anything that doesn’t throw an error but should is a bug. If an error is displayed that doesn’t accurately detail what the issue is then that is something else that should be reported. If there are any time based elements, try seeing what happens when you muck about with your phone’s settings. If it uses network connectivity, modify your settings and see how it handles those situations. Harder to create, but still worth it is testing how the app handles being interrupted by things such as phone calls, text messages, low battery notifications and other similar interruptions.
No. Absolutely not. There are plenty of other things you can test. This has simply highlighted a few of the basics that should be covered regardless of the app. If you can manipulate something in the app, test it. Test everything. Test, test and don’t forget to test.