Check test coverage: It goes without saying that TDD should be the norm and developers should pro-actively write test cases. New features should never break old ones.
Perform load testing: A must if you work at high scale.
Measure performance: This is required for you to know what kind of instances and resources you would need to boost performance.
After deployment:
Monitoring: You can use various tools for this. We use New Relic to monitor our services.
Logging: Contextual logging is very important. You need to be able to check the lifecycle of a request as it was completed.
Alerting : We use 2 kinds of alerting mechanisms
Soft Alerts: Over email or slack. These are not critical alerts but more like bringing something to your attention. Like some requests are taking 1000 - 2000 ms to respond.
Hard Alerts: You get a phone call or message about serious issues. Like server crashes, high latency (>2000 ms)