Yeah, I'd start with AWS then. Once you understand that you'll be able to map concepts to different providers.

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.