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?