What I needed was a light table. Whether you're drawing 2D animation for cel painting or doing the preliminary pencil work to be scanned in for a computer animation, a light table is one of the most important tools for frame-by-frame animation that you'll ever own. A light table (or light desk) is a very simple device--it's just a box with a clear or translucent white surface with a light inside, that shines through the surface so that any paper overlaid on it is rendered transparent/translucent (the onion-skin effect that Flash simulates). This way you can layer multiple sheets of paper on top of each other, and see your sequences of frames for reference as you're working on drawing or tweaking new frames.
Large studios or other professionals in the animation industry sometimes use highly-specialized light tables with glass surfaces over a special light; the tables can be locked, or set to rotate and pivot in order to help the animator draw with the rotating sweep of his/her full arm. Those tables are rather delicate and highly expensive, but there are much cheaper options for personal use. Any hobby shop or camera store will be able to help you pick out one of their light boxes, which are generally used for viewing slides and transparencies but that can also work well for animation as long as they have a large enough drawing surface.
Your local art store or craft shop will probably have basic light desks/light tables, though. I shop at Texas Art Supply here in Houston rather often, and that was where I picked up my ArtoGraph LightTracer for under $30. It's just the right size for an 8.5"x11" sheet of paper laid out in landscape format, and angled so that it can rest comfortably across the lap or on a desk or drafting table. The bulbs last a long time and are easy to replace.
If you really want to simplify, though, really all you need is a glass surface and a medium-strength light. I've known people to build their own light boxes with a wooden frame and a sheet-glass surface with a lamp inside, or to even just place a desk lamp underneath a glass coffee table. Whatever works for you--just don't ruin your furniture in the effort. Once you've got a light desk set up you'll find yourself animating with ease.