What is the best way to animate a 2D character?

Depends on what you want to do.

Flash and Illustrator are tools that, like Unity, try and stay out of the way as much as possible. They also aren't meant for animation. (Please don't say anything about Photoshop's animation tools because they are the spawn of the devil.) So it isn't Photoshop/Illustrator that are the problem here. I would personally, as an animator, recommend working on your animation skills. Animation is really finicky, and it can be surprising what looks good and what doesn't. It takes time to get a feel for the timing of actions and working with odd perspectives.

I'm putting my method, which runs the gamut of Adobe products but which is doable with just Flash, or Photoshop, below. It works on the idea of fast iteration. So you do minimal work before you see your animation and then iterate on it to fix/polish it.

(This is for a character but is completely adaptable for any kind of animation)

In Flash

1. Create a Movie Clip for the animation

2. Create a stick figure version of the animation

3. Use Flash's built in tools to export the Movie Clip as a PNG sequence

This way you can test your animation to make sure it is working before exporting.

In Photoshop/Illustrator (Or you can continue in Flash, skipping #1)

1. Assemble the PNG sequence in a grid.

2. Begin to paint over each frame of the stick figure animation (which you know works)

3. Save and Import to Unity.

4. Put the animation into the scene so you can see it.

5. As you keep working on the animation in photoshop, reload it in Unity so you can watch the changes.

Hope this helps! Best, Peter