Me and my friend have decided to team up and make games! He's the art man and I am the creator (for lack of a better word). What game engine would you recommend for our purposes? But now that I think about it I also do want to make other games that are not casual and for other platforms, I don't want to limit myself but the topic title was our initial thoughts. Do most people use something different than what would be recommended to us?

Unity. It can build for mobile and html5/webgl and it is quite user friendly.

Didn't check Unity recently but I seriously doubt you'd enjoy tens of MBs of webGL port (if it is still so)

Unity is a good choice for mobile but when it comes to browser,

Phaser is popular for 2D whilst three.js/babylon.js is for 3D.

You can use engines like phaser with a wrapper (cocoon etc) at mobile but it is something you should decide based on what you want and scope.

Cocos2d-JS can go for both web and mobile native. Use Three.JS for 3D, and wrap it with Cordova + Crosswalk for mobile native. They are all cheap in file size and fully JavaScript based, which means you can use the web APIs with ease.

I think Emscripten (http://emscripten.org) along with the excellent LazyFoo tutorials ( http://lazyfoo.net/SDL_tutorials/) is a really good choice for the web. Purely because there is so many great resources for C++ and SDL whereas I find the docs and tutorials for javascript game libraries a little thin (also a lot of help on forums immediately jump to using JQuery or a large framework which unfortunately isn't ideal for games).