The only thing I can think of that does it is Flash (if that even still does it), and that of course isn't free. Ya might be able to find someone with it that'd be willing to help you out though.

http://plasticanimationpaper.dk/ - Traditional doodlery, has some layers and onion skinning, The interface is a bit different
but not difficult in the end. You can't colour in this as far as I know, but you can export your frames as images and do
that separately.

http://www.pencil-animation.org/ - I've made a few in this. You have onion skinning and layers for sketching, vectors and
painting/colouring. It can export flash files directly. Some tools have issues, like when I try to move a selected piece,
it duplicates it on the sketch, and I don't know how to work around it.

http://www.synfig.org/cms/ - This is some supposedly flash-like thing, which features vectors and tweening. You can make
nice stuff in it, if you can nail down how it works.

http://www.humanbalance.net/gale/us/ - This is very nice if you want to make sprite animations, or pixel art be it a game
or an avatar icon. Has onion skinning and layers as well. Similar to MSPaint and a bit more. You can paste your
design in it to pixelate over.

I think GIMP also has some animation package, but I have never tried it.

In case there's confusion, onion skinning means you can faintly see the previous frame under the current frame. It helps to draw the next one.