We live in an older home (1940) that we purchased with 2 additions.  First addition in 1992 is approx 625 sq ft living room with 6" concrete floating floor.  Below living area is 2 car garage/walkout basement with I-Beams and posts.  There is no insulation below or above.  Living area is a 9' ceiling.  There is a vapor barrier/underlayment with laminate flooring on top. Cold radiates from the floor and is very uncomfortable and I'm sure inefficient.  This space has it's own heating/cooling unit that literally never turns off in the winter time.  My question is what is the best way to insulate this?  We've had several bids... From installing rigid foam insulation on top > subfloor > then flooring..  Installing rigid foam on ceiling of basement.. To framing the floor with 2x4 or 2x6 then spray foaming.

Do you have a ZIP code or other location information you can share? What makes sense in some climates may not in others.

Some pictures of what the garage ceiling looks like might be relevant too.

Zip code is 62002.  I attached a pic, could not attach a good quality one.  The ceiling has small layer of some sort of fiber bonded to it.

lj

So, just outside St. Louis.

You already have an internal vapor barrier.  So spray foam is kind of a no-no.
Especially since you have obvious signs that the concrete is still actively drying to the exterior.
Depending on the thickness of the concrete, maybe a couple layers of Rockwool Comfort-Board.

If you want to cover it with an air barrier, make sure whatever you use is vapor-open.

The concrete is 6". Would i add this rockwool to the ceiling or go over the concrete in the living space?

I can tell you what I did in a similar situation. I used some eps that I found on Craigslist and glued it to the ceiling with gun foam. Used 2x's for supports until the foam dried and then put another layer on going in the other direction. Simple especially with concrete. Be sure to foam the seams where they butt together. You'll have a seamless insulation system.