A short list of tools to work across browsers, devices, and platforms.

CSS+JS Frameworks:
* Bootstrap (http://getbootstrap.com): A front-end framework whose UI components work across all browsers and devices. They have an extensive plug-in library.
* Foundation (http://foundation.zurb.com): Bootstrap's main competitor. They have a different grid system for placing components.

Libraries:
* jQuery (https://jquery.com): The defining library for interacting with the DOM across browsers and devices.
* Lodash (https://lodash.com): A utility belt to work with modern Javascript even if the browser doesn't support it.
* Modernizr (https://modernizr.com): A tool to detect browser capabilities rather than check for a specific browser.

Pure JS Frameworks:
* React (https://facebook.github.io/react): Technically a library. Facebook built React to simplify UI building and provide a virtualization layer over the DOM.
* Backbone (http://backbonejs.org): An MVC framework. It's barebones and malleable.
* Angular (https://angularjs.org): An MVC framework that integrates through directives embedded in your HTML.
* Ember (http://emberjs.com): Yet another MVC framework.