What service(s) do you recommend for running Simple full-stack app? What costs can we expect?

Hi, as i understand you have a docker container for the backend with the business logic, accessing Postgres, and providing some kind of API (REST, GraphQL, etc.) to the frontend. For the frontend itself, you also have docker container which just serves the static assets to the client. Just Client Side Rendering or also Server Side Rendering?

If you do not know the traffic to the site yet, or expect fluctuating requests, and want to keep costs low, i would suggest to look into AWS Lambda for your backend business logic, API Gateway or AWS AppSync for the API Layer, and Aurora Serverless v2 for Postgres, and Amazon Cloudfront as the CDN serving the static assets directly from S3. There is also a quite generous Free Tier for most of the mentioned services. This setup scales very well, performance wise and operation wise.

An alternative approach could be AWS Amplify Hosting, which also comes with an integrated CI/CD Pipeline.

Here are some sample architecture patterns. Some parts shown are optional which you may not need, for example the Lambda Authorizer, Route53, Certificate Manager, Cognito, ElastiCache, and instead of DynamoDB you can use Aurora Serverless Postgres for instance.

I think the "Single Page Application" is closest to what you are asking for: