How to find bugs in a website ?

As a tester you should perform or consider below mentioned test scenario for web testing.

1.Functionality testing :- This should cover the functionality offered by your website.

2.Usability Testing :- Usability testing is a technique used in user-centered interaction design to evaluate a product by testing it on users. 

3.Compatibility Testing :- Issues like cross browser compatibility can be tested.

4.Interface Testing :- In case system is interfacing with external systems them interface testing helps to ensure proper handshaking.

5.Security Testing :- Security testing is a process to determine that an information system protects data and maintains functionality as intended.

6.Performance Testing:-Load and Stress testing

Enogh information is available if you use google for more information.