Parallel Development and Live Site Management

I have a web project that has the following requirements/features:

• Front end in HTML CSS and JS/JQuery
• Back end in PHP with a MySQL database.
• Uses external resources via an API (Stripe in this case, who have parallel test and live interfaces).
• Development is currently done in Netbeans
• Source Control is managed through Netbeans into a Git repository on VS Team Services online, which we are also using for development on our Windows product.
• The live site is hosted on a Linux server at a shared hosting provider, files are transferred via FTPES. When the product demand grows we will transfer the project to a dedicated off-site server.

Once we go live we need to have a reliable, manageable way of continuing development in isolation, then replicating the changes to the production area once the changes are tested and validated.

What are the tools, methods and/or best practices available for managing parallel development and live website/web-app projects?