To migrate from an AWS RDS for PostgreSQL database to a high-performance, cost-effective, and predictable PostgreSQL database with Timescale, developers can use the live migration solution provided by Timescale. This process involves several steps including assessing the current database size, complexity, and dependencies, planning a migration strategy, enabling hypertables on time-series tables, migrating data from the old database to the new one with minimal downtime, testing the new database with applications, and cutovering to the target database. The recommended approach is to use an intermediate machine to execute the migration process, utilizing tools like pg_dump/pg_restore and PostgreSQL logical decoding for a seamless transition. By following this playbook, developers can achieve 44% faster data ingestion, up to 350x faster queries, and 95% storage savings compared to RDS.