Practice your technique on non-animated pixels first. You have a few places where the branch changes the color of the outline on different frames in your animation. You will have a harder time implementing consistent changes to your entire piece when you throw animation into the mix.

First thing to notice in your picture is that you have black outlines on the trees, on the branch (except a few frames), and the cloud. Try removing all the outlines, or making them a much closer color to the item itself. (I recommend removing the outlines entirely for a piece this scale)

Next notice that, with the exception of the moon, all the colors in your piece are very close in shade and hue. This makes them difficult to distinguish to start with, and on top of that, it would make any type of AA even more difficult to notice. Also note that your bird is nearly the same color (or the same, I'm not sampling) as your background, that is going to make the outline even harder to hide while making sure it can be seen.

I would recommend choosing a (very) slightly lighter color for the sky, and switch the bird to a very dark color, almost black. Then when you go to AA, you can choose an in-between color for smoothing out the edges.

Lastly, the only place in your image that is even close to AA (in-between colors to hide the lines) is on the top edge of the moon. However, you run into two problems. First is that you have a solid line across the entire edge, which is a prime example of banding. Second, your color on the edge of the moon is much to bright. Any pixels you use for AA should appear to be neither the moon, nor the sky, but rather a color that is a bit of both.

Now there are more advanced ways to use color with cross ramps and making some great AA that way, but if you're starting to pixel those effects, feel free to play with the colors and get the concept before worrying about consolidating it.

I'm sorry I don't have any examples or edits prepared for you at the moment. I had a bit of time to write this, and now I'm being called away. I'll try to stop by again later and check back.