ASP.NET is a free web framework for building great Web sites and Web applications using HTML, CSS, and JavaScript.
ASP.NET Web API is a framework that makes it easy to build HTTP services that reach a broad range of clients, including browsers and mobile devices. ASP.NET Web API is an ideal platform for building RESTful applications on the .NET Framework.
NuGet is a free Visual Studio extension that makes it easy to add, remove, and update libraries and tools in Visual Studio projects.
You can easily find a web hosting company that offers the right mix of features and price for your applications.