I personally use and love Android Studio for Android work. I really like App Code (by the same company as Android Studio), but I admittedly have not used it enough to know if I like it more.
Jet Brains, the maker both of Android Studio and App Code, has a pretty good open source policy. If you are a maintainer/contributor to an open source project, you can apply for a free license.
Android Studio is free for anyone, and I get App Code for free via my open source license.
As native vs mobile app, I would recommend doing native apps. By default, you have access to the phone's functions, it tends to provide a better UX, and once you get over the learning curve of language/framework, I find it to be faster/easier. The last point may depend on the app, and what you already have built. e.g. An app that has no dependency on a backend server might be faster to do native, whereas an app communicating to a server may end up being faster to do as a mobile site. As a general rule, my experience is that users tend to prefer the native apps anyway.
A big -1 to phonegap. An experiment at our company lead us to 8 weeks to build the phonegap version and 6 weeks to build both native versions.
Mobile Backgound: Professional software developer for ~6 years, maintainer of iOS an open source project, have given around 6 talks at various iOS conferences.