SSMS works against Azure databases, so you can use that.

As far as migration, you connect with TDS and then move the data. Red Gate's Data Compare will work, as will SSIS, or most SQL tools.

Why move the databases to Azure? There are ongoing costs involved with keeping things there, and you might need to rethink your app. I think Azure works well when you architect something new, but if you have the hardware and space, why move?

Disclosure: I work for Red Gate Software