How I should make Integration tests for a Rest client on a web api?

I need to make integration tests using xUnit for a Service that I made mainly making REST calls to a Jenkins host.

Since the Jenkins web api is not made by me, I know that I don't have to test what is already provided as native features.

The question is, what should I test single methods, result of native features with a given input, bad status code on bad input, resources creation ?