AWS offers many services, but for me one of the best is the RDS. It saves a lot of time and makes my development happy - I don't have to worry about data loss or fancy log rotation techniques. If you remember the times when database backup was created by custom scripts, you will surely like that RDS will do it all for you. It is not only about automation or excessive use of the cloud. It's more about reliability, stability and managed services - this is what you pay for.
The following scenario describes my use case of using AWS RDS. As of today I use it for 7 web applications. In total all web apps serve data for about 1000 unique users daily. Each app is cache-optimized (with nginx pagespeed and redis). Some prices can be different in each region, but key parts are same. Okay then, what to choose? With AWS RDS, you mainly pay for two things: Service and Storage. The good news is that both costs can be optimized.
Reserved Instance
To optimize Service costs you have to purchase the Reserved Instance. My choice is one instance of db.t3.small (2 vCPUs, 2GiB RAM) without Multi-AZ and All Upfront payment for a one year. The database is PostgreSQL. The db.t3.small is the cheapest database with at least 2 GiB of RAM - which is suggested in PostgreSQL documentation as a minimum hardware requirement.
Storage
Once you purchased the Reserved Instance you have to run the RDS instance. Select same options as you selected in Reserved Instance purchase form. And here is the game changer, you must update the Storage type by default it is set to IOPS SSD (but it's pretty expensive ~$238/1000 IOPS/month). Change it to GP2 (General purpose) the price is going down to something about $5/month for 50 GiB of disk space.
Summary
By doing the above, you can end up with one-time cost per service like ~$240 per year and monthly $3-10. Bear in mind that everything depends on region, disk space, instance class, traffic and the application itself. You always need to know what you have to pay for in the new-cloud-world. If you want to run your side projects and need long term database service then AWS Reserved Instances with GP2 is the best option to start.
AWS also offers the calculator where you can estimate your costs, here is the link.