Some things to consider:

• This is also a good time to make your house well-insulated, like a "passive house". To ensue good air quality when all the windows are closed, you'll want an air filtration and ventilation system, ideally ductwork that cycles air through the house. Even if you weren't planning on the insulation part of things, the ventilation strategy required for that kind of design promotes the highest air quality because it tackles the problem of, "how do we keep the air filtered, circulated, and dehumididied?"

• If this is too expensive, you can implement the basic parts of the strategy on a per-room basis. Ventilate to/from the outdoors with fan systems with replaceable filters. Use a dehumidifier system. Keep your heating system clean or filtered so that heat isn't burning dust.

• Don't use natural gas for anything. It doesn't burn completely and the byproducts are bad for air quality.

• The worst air quality rooms are bathrooms if humidity isn't controlled (leading to mold) and kitchens because there is inevitably combustion there even without gas. It's good to have windows and a very powerful fan system by your stove. Something much more powerful than the usual hoods. Like the kind people use for home cooking in woks. Ideally cook outside as much as you can. If you can add a semi-outdoor kitchen that's the best.

• Get the best wall siding (e.g. sheetrock) for your climate. Certain kinds are better against mold than others.

• If you have any existing ventilation systems that you plan on keeping use this opportunity to clean them and put filters on them so you don't need to clean them again.

If the kitchen has a gas stove then you should install a ventilation hood that vents to the outside, that's probably the biggest air quality win you could make. (Edit: Well, aside from switching to induction, if that's in your budget)

I dunno about filters that can fit in wall cavities, but for general filtration: "The Best Air Purifier Is A $150 DIY Option". I suppose you can DIY something that saves some space by tucking into a wall cavity, but I don't know if it's worth sacrificing the insulation, unless you're also building ventilation to the outside using the same wall. I would poke around DIY/workshop youtube channels, a quick searched turned up this build that utilizes ceiling space. (At least with the walls down, you can more easily install ceiling outlets.

I use an older AirGradient to track particulates and CO2, it's been great for knowing when the room I'm sitting in is spiking in CO2, so I can crack a window or something.

Good insulation is key if you can do that. Also, look into heat exchangers for air, there are some DIY versions with filters that can give you good energy efficiency while allowing you to filter the air coming out of the street without needing to open the windows.