Actually for your questions, you should know what is Azure SQL database and it's capabilities.
- I'm leaning towards using Azure SQL DB for multiple reasons (products offered, transparent pricing, integration ease, available documentation, SSO, etc.), but want to first understand the skills needed and ease in maintaining it in the long run.
This document What is Azure SQL Database service introduced almost all message you want to know. SQL Database is a general-purpose relational database managed service in Microsoft Azure that supports structures such as relational data, JSON, spatial, and XML. SQL Database delivers dynamically scalable performance within two different purchasing models: a vCore-based purchasing model and a DTU-based purchasing model. SQL Database also provides options such as columnstore indexes for extreme analytic analysis and reporting, and in-memory OLTP for extreme transactional processing. Microsoft handles all patching and updating of the SQL code base seamlessly and abstracts away all management of the underlying infrastructure.
- Will we have to manually rebuild indexes and partition out tables as we scale up? Or is Azure intelligent enough that it'll do most of the heavy lifting of performance optimization itself?
No, you don't. Scalability is one of the most important characteristics of PaaS that enables you to dynamically add more resources to your service when needed. Azure SQL Database enables you to easily change resources (CPU power, memory, IO throughput, and storage) allocated to your databases.
You can mitigate performance issues due to increased usage of your application that cannot be fixed using indexing or query rewrite methods. Adding more resources enables you to quickly react when your database hits the current resource limits and needs more power to handle the incoming workload. Azure SQL Database also enables you to scale-down the resources when they are not needed to lower the cost.
For more details, please reference: Scale Up/Down.
- Does Azure or other vendors provide services to optimize a DB?
As Woblli said, Azure SQL database provides the Azure SQL database Monitoring and tuning for you.
As a complement, you also can use Azure SQL Database Automatic tuning to help you optimize the database automatically.
Hope this helps.