Yoren Chang has written a useful guide to the basics of using Angular in WordPress theme development, and a set of more in-depth articles about Angular and the WordPress API.
React, which was created by Facebook, is a much simpler project than Angular. It’s been described as the V in MVC and is essentially concerned with managing the display of data on the front-end and building interactive user interfaces. Whether you think of that as a good thing depends on the complexity of the application you want to build on top of WordPress and whether you’d rather do server or client-side rendering.
For simpler integrations with the REST API, React is probably a better choice than Angular. React is particularly notable for its use of the Virtual DOM — changes made to the view are made to a Virtual DOM and the actual DOM is only updated once those changes are completed. The result is a fast and fluid user experience.
Take a look at the React homepage for some examples of React code. Kelly Dwan has created a simple WordPress recipe theme using React, the GitHub repo of which is a great way to learn about how React can be used with the WordPress API.
Backbone is a full MVC framework, but it’s both smaller and less complex than Angular. Backbone doesn’t have the buzz of the other solutions I’ve discussed, but it’s has been used on some serious online services, including Twitter, Pinterest, and Disqus.