You need to perform below mentioned test in order to find bugs in a website:
- Performance Testing - To check if the application is able to cope up with excessive load. Page loading speed should be good.
- Security Testing - To prevent your website from being hacked you need to perform the security tests. This is mandatory if your website is offering facilities like banking, shopping, social networking etc.
- HTML and CSS validation - There are many tools like HTML tidy, Google Webmaster tools etc. that can help you in validating your HTML. CSS Compressor is a tool for helping you out with CSS.
- Cross Browser Testing - It is vital to check whether your website is supported through multiple browsers as not all web elements like AJAX, java applets, etc will be rendered in different rendering engines. Simply, put a website working well in Chrome may or may not work well in Mozilla Firefox, IE, Opera or any other browser.
- Mobile Ready Test - In this mobile-first world, it is essential to create a website that is mobile ready. So you need to check how responsive you website is through different mobile devices.
- Accessibility Testing - You need to test if your website is accessible to all including people with disabilities. There is a set of guidelines issued by W3C which must be followed for every website before launching it in the web.