Next.js 13 has adopted the App Router as its recommended way to handle routing, leveraging React's latest features for optimized speed and performance. To set up application performance monitoring with New Relic, install the agent using npm, modify the next.config.js file, and add configuration files such as newrelic.js and .env. The integration provides comprehensive observability of server-side activities, while also allowing client-side telemetry through the injection of the New Relic browser agent. To capture detailed error information, handle both client-side and server-side errors using code samples provided in the blog post, which can be found in the newrelic-node-examples GitHub repository.