You definitely should have unit tests for your apps. Whether you follow TDD strictly or not these can be very useful, especially for regression tests. Most continuous integration tools out there fully support running unit tests. Just don't try to cover everything, focus on the critical parts of your app and if time allows, then proceed to the other parts. It is unreasonable to think that we will have 100% coverage in most projects, so we need to make decisions. Mocking frameworks play an essential role here as they allow us to simulate third-party services nicely.
Automated integration tests as we saw here allow us to test features that aren't available in unit tests, and these cover another part of our needs.