Testing for LWC is implemented in Jest. These can be run locally to your org, or even enabled to run in "watch" mode to give you instant feedback while you code.
From the docs, LWC testing enables you to
- Test a component in isolation
- Test a component's public API
- Test basic user interaction with a component
- Verify the DOM output of a component
- Verify that events fire when expected
There are a number of tests implemented in the e-bikes sample app in the sample app gallery.
You can also read about it in the developer guide at https://developer.salesforce.com/docs/component-library/documentation/lwc/lwc.testing.
Spring 19 Pre-Release Note
Until it is updated for prerelease, you'll have to login to a pre-release org, and then go to the url: