Deploying containerized Docker instances in production?

Hello! After spending many development hours in my past years running on Virtualbox/Vagrant-style setups, I've decided to take the plunge into learning Docker, and after getting a few containers working, I'm now looking to figure out how to deploy this to production. I'm not a DevOps or infrastructure guy, my bread and butter is software, and although I've become significantly better at deploying & provisioning Linux VPS's, I'm still not entirely confident in my ability to deploy & manage such systems at scale and in production. But, I am now close to running my own business, so these requirements are suddenly going from "nice to have" to "critical".

As I mentioned, in the past when I've previously developed applications that have been pushed onto the web, I've tended to develop on my local machine, often with no specific configuration environment. If I did use an environment, it'd often be a Vagrant VM instance. From here, I'd push to GitHub, then from my VPS, pull down the changes, run any deployment scripts (recompile, restart nginx, etc), and I'm done.

I guess what I'm after with Docker is something that's more consistent between dev, testing, & prod, and is also more hands off in the deployment process. Yet, what I'm currently developing still does have differing configuration needs between dev and prod. For example, I'd like to use a hosted DB solution such as DigitalOcean Managed Databases in production, yet I'm totally fine using a Docker container for MySQL for local development. Is something like this possible? Does anyone have any recommendations around how to accomplish this, any do's and dont's, or any catches that are worth mentioning?

How about automating deployment from GitHub to production? I've never touched any CI/CD tools in my life, yet I know it's a hugely important part of the process when dealing with software in production, especially software that has clients dependent on it to function. Does anything specifically work well with Docker? Or GitHub? Ideally I want to be avoiding manual processes where I have to ssh in, and pull down the latest changes, half-remembering the commands I need to write to recompile and run the application again.