I am wanting to create my own online SQL database for a website that is going to feature some interactive r shiny/plotly apps. I see that there a appear to be a few options through things like amazon web services and microsoft. Does anyone have any recommendations for something that would be free and minimal work?
I can recommend you to try AWS and/or Google Compute Engine.
AWS has free tier plans that include everything an operating system can do, but the free tier plan is limited both in time and functinality.
Gearhost also offers free plans that include a database, but the offer is limited. It can make a large difference if you only pay $5 monthly, then you'll be able to use more advnced option more suitable for production.
If your project is not for production and only development, you can go with the free tier plans.
AWS may be good to get your feet wet, but once your free trial is up, it is MUCH more expensive than Linode.com for the resources you can get.
Additionally, with Linode, you have complete and total control of your VPS - what Linux (or BSD) distribution you run, you are the root user, you are in control of all of it.
Linode.com is $5/mo for the cheapest "machine" you can up or down size on the fly at any time, they do hourly billing or by the month, great company to work with, been a happy customer for 12+ years.
When you sign up, use promo code DOCS10 for a $10 credit.