They go hand in hand. If you write tests, but they're not run after each dev commits changes, you're not testing those changes for regressions. If you have CI without tests, what is the CI going to do? It could still be used for packaging executables/distributions, and possibly orchestrating deployments, but it sounds like that's not your goal.
Save yourself some grief and use a service like CircleCI, it provides isolated (Docker) environments that are spun up per test run. It supports many languages and background services (like databases).
As for writing tests, you would be best to start writing them at a "system boundary". This might take the form of sending HTTP requests to your endpoints and parsing the response.