Animations can easily be saved as Movie Clip or graphic symbols by copying and pasting their frames.

You cannot simply select an animation (series of frames) and convert to symbol. It will only take one frame, the frame that you are on.

As a work around for this:

1. Select the frames you want with the arrow tool by dragging over them in the timeline.

2. Select Edit>Timeline>Copy Frames

3. Create a new symbol. Insert>New Symbol

4. Select frame 1 of that symbol or wherever you want your animation to start.

5. Select Edit>Timeline>Paste Frames

That's it. It should be pasted in. Layer names may not come with it, so you may need to rename them.