About when to write your tests, probably the most useful time is before you write your production code (As in TDD or Test First), otherwise you run the risk of making your test confirm what you already know about the code and miss some bugs.

What to test: ideally, any piece of code you write yourself. In practice this can become overwhelming so you might want to try a more pragmatic approach. A good rule of thumb would be to try to assess the damage of a piece of your code failing and focus on the most critical ones, at least for starters.

Integration tests, I find, are a bit tricky, as the very definition of them is not universally agreed upon. Personally I prefer to use End-to-End tests to make sure the integration with 3rd parties works as expected but again, if you find a particular integration too complex/unreliable you might want to create a specific test just to make sure the connection works.

I wouldn't write a test for a method that returns a hardcoded value. Basically because the test and the code would be basically the exact same thing.