To clean the finish on your wood furniture just use warm water and mild dish soap. Cleaning furniture with water won't hurt it. Just be careful not to soak it. Use an old toothbrush for hard-to-get areas. Wipe the dirt off the surface with soft cloths or paper towels until it comes up fairly clean. Then dry off any moisture with a soft cloth.
After a good cleaning, the best way to protect the finish is to use a good quality soft paste wax. Apply a thin coat as directed on the label. Wait five minutes and buff lightly with a soft shoe brush or cloth. Wait another 30 to 60 minutes and buff/brush again with a bit more vigor. You'll see a beautiful shine return to the finish that will last for many months.