Where do you live. Adding the correct amount of insulation will be regionally sensitive.
The ceiling grid is attached to the walls, so it would seem the ceiling gets removed first. You can disassemble it and reuse the parts if they're in good shape, but the two foot fill-ins won't match in color unless you paint all the tracks. If you weren't doing work on the walls the ceiling track system could stay in place.
Personnally I would demo the ceilings and walls. How you handle the wall structure will depend on the amount of insulation you want.
A drywalled ceiling would cost much less then the tiles.
Ron

Ron, thank you for replying. We live in Lancaster, PA. My husband says there is insulation behind the paneling, but we want to change it out.
We have not had any leaks but how difficult is it to change out portions of a drywall ceiling?

I am thinking all of the following needs to be done first.

We need more lighting, recessed ceiling lights would probably the best choice. Also hiding some phone and computer wires, and 2-3 additional elec. outlets and adding another cable outlet. I want new mouldings. And of course new insulation.

We could do the demo work, paint the drywall and moulding, and change out the thermostat and elec. baseboard heaters.

Things like new doors, fireplace facelift, and carpet can come later.

"We have not had any leaks but how difficult is it to change out portions of a drywall ceiling?"

I've had a drywall ceiling for 15 years in the basement and haven't had any reason to touch it. If you want to identify any potential problems while the ceiling is open, fine. This way you minimize opening the ceiling at a later time.
Say for instance there's a bathroom above on the first floor in a house that's 40 years old. I might look at the tub or shower traps as potential issues and replace them while the ceiling is open.
Every house is different and you don't need to anticipate every possibility. It's drywall. If you need to cut it open, cut it open. You repair the ceiling and paint it.
It's not brain surgery, but alot of the tools are the same.
Ron

It's not brain surgery, but a lot of the tools are the same. In a documentary, I actually saw a surgeon wielding a Black & Decker drill to open a skull for brain surgery. When questioned, he said he found it a lot better than the overpriced stainless steel models that he used to use.

Given the amount of work you are proposing, it would probably be best to gut the entire room all the way down to the ceiling joists and bare concrete walls. There are special insulation considerations for masonry walls that differ from wood framed construction (check out the basements forum).

worthy and brewbeer, thanks for your replies. I am guessing we will gut the whole thing when we start. About the only items that could wait are the new carpet and fireplace facelift. But that fireplace work is so messy it would be better to do it with everything else.