Synfig Studio is a really powerful free and open-source 2D animation software for Windows, Linux and MacOS X. Although there is a certain learning curve to Synfig (you didn’t expect animation to be that easy, right?), the internal consistent logic of Synfig makes creating professional animation fast and hassle free once you understand how things work.
To help you on your way, Synfig runs regular (and cheap) courses at Udemy. Synfig can output video and GIFs and supports all major formats and screen size.