I'd start with fine steel wool with mineral spirits to see how much it would clean up with that mild solvent. I wouldn't expect the stains or water spots to disappear, just fade enough to get that well-used rustic effect. If that leaves an appearance you can live with, I'd refinish with multiple coats of Tung Oil.
JMHO

Clean it thoroughly with mild soap and water, rinse with clean water and dry immediately with a soft cloth

then clean it with mineral spirits and fine steel wool, rubbing gently. Wipe off the crud with a clean cloth.

Let it dry and see what it looks like. A light sanding on the top may be needed to minimize the damage.

To fix the split panel, remove any nails holding the panel in place (it should be set in a groove and able to freely expand and contract. Then apply wood glue in the crack and shove the sides together.

then you can apply a penetrating oil finish, a gel urethane topcoat, or just a good waxing with a hard paste wax.

The top is is bad shape and will need some work to get it looking good.

Option 1 is to apply A-B wood bleach to remove the stains.
Option 2 is to just paint it a nice satin black oil-base paint

You can glue the door panel crack together if you wish. Replace the hardware (drawer pulls)

For the sides,
Option 1 is to apply shellac, this will bring out the depth of the wood. But it's a gloss finish.

Option 2 is to apply an oil-varnish blend. Examples of this in the article below. Wipe-on ; wipe-off several coats will give a nice in-the-wood finish. About as foolproof as you can get.`

Here is a link that might be useful: Oil Finishes

Thanks for all the info! So far, I've cleaned it off with soap and water. Next step will be mineral spirits and steel wool. I may try the bleach and then reassess.

another option would be to faux finish the top to make look like marble or granite, or some other stone.

When done by someone who knows what they are doing, you really have to look close to see it's a faux finish.

This post was edited by bobsmyuncle on Wed, Aug 14, 13 at 20:35

Yeah, it's just that I don't know what I'm doing :(
So, I'll stick to your foolproof option above.

another option would be to faux finish the top to make look like marble or granite, or some other stone.

When done buy someone who knows what they are doing, you really have to look close to see it's a faux finish.

Dont know how "correct" this is (you would not do this with a fine antique), but after cleaning off the dirt and old wax,with something with this degree of wear and staining, I like to do a bit of light sanding just to get the wood smooth and to fade out the stains a bit. With the top I might get more aggressive and actually go down enough to remove more of the scratches.

But the trick Ive found is to know when to stop and not do TOO MUCH sanding. Itll never look new so dont even try - there is indeed a point where it can be ruined by too much. You want it to stilll have a lot of soul and look old and used!. So, youll still have some of the deeper scratches and dark rings, but overall the grain of the wood should clearer and more visible and the color of the wood brighter.

Is there still any finish or traces of finish on it? That can be removedafter cleaning and before sanding with furniture refinisher (NOT the same thing as paint stripper) and steel wool.