Check out Google Cloud. They have a much better developer experience than AWS.
Cloud Run is pretty easy to setup.
Google Cloud Run. One of the few good parts of GCP.
It supports containers, and has a relatively generous free tier
Have you looked into PaaS, for example, Render: https://render.com/docs/deploy-an-image
You might like https://www.flightcontrol.dev which makes it easy to deploy servers or Lambdas. (I'm cofounder)
I don’t think you will learn to deploy anything serverless in less time than you could deploy a server.
And you will learn more general devops skills instead of one particular provider’s serverless API.
It will be an express based REST API that connects to an external Database.
The API will have to expose endpoints and do CRUD operations for the most part.
Can you provide more information of what you are aiming to archive in regards of your API?
Skip the UI and use Terraform
Did you evaluate Fly.io?
Do you have to use Docker? If you can write the func in Javascript, Cloudflare Workers are amazing and so much easier to use than AWS.
You can also write them write inside Next.js and deploy them to Vercel with a single click.