Amazon Aurora is a database as a service offering with improved features and reliability compared to RDS, but also comes with complexities and cost calculation challenges. To reduce costs, it's essential to understand how the database is being used, optimize storage usage, and set up billing alarms. Strategies for reducing costs include optimizing non-production environments, using the correct type of I/O access, switching from Serverless to provisioned instances in production, right-sizing instance types, trimming backups, offloading large objects to Amazon S3, and optimizing SQL queries. Additionally, Timescale provides a cost-effective alternative for time-series data projects, offering infinite scalability storage tier with low-cost usage-based storage costs.