Some common types of web tests you could start with are content, links, function and dynamic element tests. The easiest would be content testing and hardest would be functional testing. Which type of testing to perform first could vary depending on a variety of factors including project budget, user expectation and update frequency.
This is the simplest type of testing useful for checking content on a page that does not change. For example, you could check whether the logo of your site is broken by checking the img tag.
Useful when: files are often moved to different locations
You could automate clicking through links to check if the expected page matches the actual page served. It is great way to perform a sanity check before pushing code to production.
Useful when: links are regularly changed and files ocasionally relocated.
This is the most complex type of automated test. In this type of test you specify a scenario that will test a specific function of your web app. The scenario can be though of as a step-by-step recipe that steps through multiple pages performing actions like filling forms, uploading files, interact with fields like checkboxes, dropdown boxes and any other manipulable element on the page. After the flow completes you can also check whether your expected matches the actual results by querying the database.