Also it is not 1GB or 10GB database, you can start up with 20GB if you want. It's going to cost money. Looks like it can range from 11c/hr to $3.10/hour depending on the size of the database you want to take to the clouds.
Sign up here for Amazon RDS
When you sign up you will be agreeing to use the following:
Amazon Elastic Compute Cloud
Amazon Simple Storage Service
Even if you don't use the latter two, you may have to pay for ECC since RDS uses ECC for CloudWatch Metrics.
After this you will be allowed to explore Amazon RDS
What about the tool to use to work with Amazon RDS?
You can use the MySQL tools to work with the RDS. For monitoring the usage, you use CloudWatch.