How do I clean & polish old furniture?

Try following these steps and see what kind of results you get:

https://www.bobvila.com/articles/how-to-clean-wood-furniture/

First wash with Murphy’s oil soap very well you will see the water get realy dirty change the water because you will be putting dirt back on

We got this recipe from an antique dealer to clean up our reclaimed doors:

1/3 shellac

1/3 turpentine

1/3 double boiled linseed oil.

Mix whatever quality you need, keeping to those ratios and then wipe down the wood (wear gloves) and leave to dry. Pretty strong VOCs, so work in a ventilated space.

Here are techniques to rejuvenate old furniture.... https://www.familyhandyman.com/woodworking/furniture-repair/how-to-refinish-furniture/view-all/

I like the rejuvenators from Formby's or Watco you can get at True Value or most home stores