Since you are right at the start of your career, you may as well plan on maximizing the life of the knowledge and skills you’re building. By this, I mean spend your time learning the newest and most advanced software rather than the old approach. Is there still work for people who only know SQL Server 2000? Sure. However, if you’re looking at the future, I strongly advocate for going with online, cloud-based systems. This is because, more and more, you’re going to be working with online, connected, applications. If the app is in the cloud, so should the data be. Azure and the technologies within it are absolutely the cutting edge today. Spending your limited learning time on this technology is an investment in your future.
This answer is a tougher call for me. Looking at new database developers (or development DBAs or database engineers or whatever…), I think the case is pretty solid: there’s so much skill overlap that it’s relatively easy to move from Azure SQL Database to on-prem. With production DBAs, the story’s a little different: as Grant mentions, this is a Platform as a Service technology, and so the management interface is going to be different. There are quite a few commonalities (common DMVs, some common functionality), but Grant gives a good example of something which is quite different between the PaaS offering and the on-prem offering: database backup and restoration. I think the amount of skills transfer is lower, and so the question becomes whether the marginal value of learning PaaS before IaaS/on-prem is high enough. Given my (likely biased) discussions of Azure SQL Database implementations at companies, I’d stick with learning on-prem first because you’re much more likely to find a company with an on-prem SQL Server installation than an Azure SQL Database.