There are two main concerns with using animated gifs in WordPress: dimensions and file size.
When you upload a gif file to the media library, WordPress stores the original file and additional files for different sizes ( medium, thumbnail ). Gifs of other sizes will be stored with one frame of animation only, so they will lose their animation capability. While adding an animated gif into a post/page, always select the full size ( this is the only one which preserves animation ).
Animated gifs could be very large. It is important to compress them before uploading to the media library. To compress them without losing quality, you should use a program employing lossless compression algorithm. You can find a good one, called Gifsicle on GitHub. This is a command line tool, so if you are not comfortable with command line, you can find other programs offering GUI. For Mac – ImageOptim, for Windows – FileOptimizer. There is also a web application Compressor.io available.