How do I remove all the grime from this table?

Plastic scrape like these to gently remove excess.

https://www.amazon.com/gp/product/B07F5XMBKW/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1

Then mix a paste of baking soda and hydrogen peroxide to scrub with a toothbrush.

Since you're thinking this is probably furniture polish and age, I would try something else. Put the table on a drop cloth and mix about a quart of warm water with a tablespoon of Dawn dish washing liquid. Stir to make sure the soap is evenly distributed and then, using a dish scrubby, gently rub the soapy mixture on the surface. You can work in sections or whatever. Wipe away any big clumps that come up as you work. Once you're happy with the result, use a rag, soaked in clean warm water, to rinse away any leftover soap and residue.

Denatured alcohol! I got that tip from a person that re-purposes wood furniture for a living.

I've found that usually a good grease cutting cleaner will cut it right out.

Unless you intend to refinish it, I'd try Murphy's Oil Soap first, because it won't harm the wood.

From the manufacturer:

"Economical, natural cleaner is ideal for mopping wood floors, paneling, and other large wood surfaces. Safely cleans finished wood furniture, cabinets, and floors. Murphy Oil Soap removes wax build-up while leaving no residue behind."

You can buy it most anywhere.

Hi Cindy I like Formby's products they make Formby's Build-Up Remover that states:

Wood furniture treatment cleans away dirt and gently dissolves wax build-up from wood surfaces.

Hope that helps, good luck with your project!!