Since the original answer was written back in 2012, let me update my current preferred 2015 client side stack:

• Twitter Bootstrap (http://getbootstrap.com/)
  HTML/CSS/JS framework with nice looking default styles and standardized HTML structures
• Font Awesome (http://fontawesome.io/)
  Free font-based icon framework
• jQuery (http://jquery.com/)
  Most widely used DOM library
  
• Require.js (http://requirejs.org/)
  Requirement handling & lazy loading for JS files
• Twitter Flight (https://github.com/flightjs/flight)
  A component-based, event-driven JavaScript framework

As a dessert for those quickly built sites you can load Bootstrap & Font Awesome directly from Bootstrap CDN (http://www.bootstrapcdn.com/)

Lets take a quick round around the track and see the essentials for any client-side athlete. The equipment depends on your goal range.

STARTING LINE:
Palms sweating, heart pumping. Ready for another race. Go!

• HTML5Boilerplate (http://html5boilerplate.com/)
  Normalized starting point for any web application

FIRST CORNER:
Accelerating into the first corner

• Modernizr (http://modernizr.com/)
  Use HTML5 elements today
• jQuery (http://jquery.com/)
  Most widely used DOM library

(Happily, HTML5Boilerplate gives you all this out from the box, so you'll get a headstart. Winning!)

BACK STRAIGHT:
Making good headway, just don't stop now!

• Respond.js (https://github.com/scottjehl/Respond) / css3-mediaqueries-js (http://code.google.com/p/css3-mediaqueries-js/)
  Media query polyfills for responsive design
• Also, extend jQuery as you go with plugins (http://www.google.com/#q=jquery+plugins)

LAST CORNER:
Time to show what you've got!

• Handlebars.js (http://handlebarsjs.com/)
  Nifty template engine for those single page applications
• Underscore.js (http://underscorejs.org/)
  Utility library

FINISH LINE:
Aiming for the big boys. Give everything you have!

• Require.js (http://requirejs.org/)
  Module loading
  
• Backbone.js (http://backbonejs.org/)
  For MVC structures in or between client- and server side

VICTORY LAP:
Fistpump for victory. Also, check out these goodies.

• Twitter Bootstrap (http://twitter.github.com/bootstrap/)
  Development framework with nice defaults
• Cross-browser polyfills (https://github.com/Modernizr/Modernizr/wiki/HTML5-Cross-Browser-Polyfills)
  Polyfills for adding functionality to older browsers