After doing some searching online, it looks like paint.net is a common
tool people use to create and animate sprites. Here's a small tutorial
on how to get some basic sprite sheet testing running.

Installing paint.net

To install it, simply go to http://www.getpaint.net/ (be careful about
others sites you go to, I end up at a couple of malware sites before I
arrived at the proper one).

The direct download URL
is http://www.dotpdn.com/files/Paint.NET.3.5.10.Install.zip. If that
doesn't exist, try to navigate to getpaint.net and click download, and
then "download paint.net 3.5". Be careful! There's a lot of adword links
that look like download pages.

Downloading sprite animation plugins

Next, you need to download the plugin! There's a couple of sprite sheet
plugins out there. I chose the spritesheet animation plugin:

Spritesheet animation plugin

Just download the "PDNSpriteSheetAnimation.zip", which will contain
a dll file. Drag and drop that into C:\Program
Files\Paint.Net\Effects. C: is of course, interchangeable with
whatever drive you installed paint.net on.

After you've installed the plugin, restart paint.net. It is now
available via:

Effects > Animations > SpriteSheetAnim :

Image

And there you go! The plugin opens up and you can test various
animations with it.

Credit for the sprite sheet in the picture goes to 10firstgame's
wordpress blog
.