Deploying the app is not complicated - just a bunch of cloud instances and object storage.
What is complicated is interlinking regions/zones, understanding current other offerings (things like lambda, serverless offerings, strategies to simplify/reduce egress, etc)
I wouldn't focus on other offerings (lambdas etc) until you are solid on fundamentals (the regions/zones and networking stuff) that allow you to deploy the current product.
I'm most familiar with AWS (used to teach certifications for them) and think you'd benefit from studying for the AWS solution arch - associate exam. It focuses on core cloud concepts and AWS. Given the customer distribution, that's where I'd start.