Best way to build interactive mobile web app

We’re trying to build a mobile first web app with wordpress. Our design is very heavy on same page interaction (swipe, tap to show an overlay window, tap to expand, etc), and less about linking to a different page on each click.

What’s the best approach to develop this kind of web app? Is there any theme/plugin that specializes in this kind of design? If we’re planning to create a new theme, is there a good starter theme for this kind of web app?