AWS has a free tier that is available for one year after you sign up. However the instance sizes in this free tier would not run SQL Server. You can build instances out in AWS and for probalby $20 a month play with what you need. Azure offers the SQL Server Developer Edition in their marketplace so you pay much less for this server. If you go through Visual Studio Online and sign up there is a $25 USD/month subscript offered. You can likely use this to play with what you want in SQL Server, just don't leave the server running...and remember the larger the server you pick the more you pay per hour. As far as learning SQL Server, there are other questions already on this QA site and blogged about that cover how you do that.
it would be good if you have a basic familiarity with core AWS services like Simple Storage Service (S3), Elastic Compute Cloud (EC2), Virtual Private Cloud (VPC) and Identity and Access Management (IAM).