Once your service is built and deployed to Cloud Run, you'll be able to monitor logs streaming from the deployment of your service. Once the deployment completes, streaming stops and you'll find two URLs, one to your live service and one to your service logs, in your output window.
By following the URL, you can view a detailed log of your service and filter entries based on their severity level. Additionally, you can monitor the status and traffic of your service through the Revisions and Metrics tabs, and adjust access rights with the Permissions tab.