Is it really necessary to test a web application? 

No matter how big or small your web app is, it’s important to at least examine your app, because special cases may get your app to malfunction.

First of all you should use Unit testing to test every method in your application. Make sure that every method is meant to do what it actually suppose to do. Check for handling exceptions properly.

Unit testing is a method by which individual units of source code, sets of one or more computer program modules together with associated control data, usage procedures, and operating procedures, are tested to determine if they are fit for use.

Once you have done the Unit testing proceed to White-box testing.

White-box testing (also known as clear box testing, glass box testing, transparent box testing, and structural testing) is a method of testing software that tests internal structures or workings of an application, as opposed to its functionality. While white-box testing can be applied at the unit, integration and system levels of the software testing process, it is usually done at the unit level. 

Then proceed to Black-box testing.

Black-box testing is a method of software testing that tests the functionality of an application as opposed to its internal structures or workings. This method of test can be applied to all levels of software testing: unit, integration, system and acceptance. It typically comprises most if not all testing at higher levels, but can also dominate unit testing as well.

Next step – Integration Testing.

Integration testing (sometimes called Integration and Testing, abbreviated “I&T”) is the phase in software testing in which individual software modules are combined and tested as a group. Integration testing takes as its input modules that have been unit tested, groups them in larger aggregates and applies tests defined in an integration test plan to those aggregates.

Then test the UI and make sure that all GUI elements are working fine.