The RStudio tutorials is well done, and includes video-flavoured and text-flavoured presentations.
You can deploy public apps to the cloud via integrated service shinyapps.io.
Containerized apps are a convenient way to deploy code in general and to package up an interactive data analysis in particular. The dominant toolchain for this purpose seems to be Docker. R goes well in Docker. Ergo, deploying r apps to Docker seems a low lift way of doing it. Juan Camilo Orduz teaches us how to Dockerize a Shiny App, as does Paul Stevenson and there are shiny server dockerfiles. The sevenbridges dockerfiles demonstrate some sophisticated configuration.