How do I clean old wood furniture?

To remove polish steep two tea bags in boiling water. Let the tea cool to room temperature, take a soft cloth, wring it out in the tea until it’s damp and wash the wood. The tannic acid from the tea is wonderful for maintaining wood.

You can rub out surface scratches in your wooden furniture with a paste made of mineral oil and pumice (available in powder form at hardware and paint stores). Use extra-fine-grade steel wool to rub the mixture into the scratched area. Then wipe it off and buff with a dry cloth.

There is a cleaner at the dollar store called Awesome. It removes wax and grim buildup. It's super effective! I also use Krud Kutter (found at Home Depot or Lowes) to clean furniture really good.

Murphys oil soap works best

My vote is for Murphy's Oil Soap, then Old English products to condition and shine.

TSP for the stubborn grime, https://www.doityourself.com/stry/best-ways-to-use-tsp-cleaner