"For a house built on a slab, it looks to me like a vent is drilled into the slab and then a fan draws the bad air out from under the slab, sucks it out of the soil, it goes up the vent and out the roof? "

That is about it.

You depressurize under the slab (a few inches of water static pressure)and exhaust away from the ground and windows to the exterior.

"alternatively is there a way to tie this in with a ventilation system for the whole house to improve air quality? "

Generally no.

You want to get rid of the radon containing air with as little chance of allowing it to enter the building as possible.
Bringing it anywhere near HVAC air is not a good idea.

"For a house built on a slab, it looks to me like a vent is drilled into the slab and then a fan draws the bad air out from under the slab, sucks it out of the soil, it goes up the vent and out the roof?"

Close. Radon gas is under pressure from the strata underneath it. The opening in the slab and the pipe provide the avenue of travel for the gas. The fan creates a small vacuum at the opening in the slab and draws the radon faster.

The slab has to be airtight for all that to work. Sump pump cavities have to be sealed, drains air tight and any points of entry through the slab(plumbing/electrical/etc) have to be sealed. Any cracks need to be sealed in case they go all the way through the slab.

Radon is heavier than air, so what comes out of the stand pipe on the roof will fall towards the ground---which is why brickeyee specified the exhaust not be above HVAC/doors/windows.

Radon abatement is basically a single purpose system.

If you live in an area known for radon concentration as I do, it's a sure bet that any homebuyer will want a system in place or deduct the cost of one from the asking price.

IMO, this is not beyond DIY; the previous responses cover the req's and a detail oriented person could achieve them. Note that the purpose is not to eliminate the gas but rather prevent concentration.

A buyer may look askance at DIY...there are third party testing labs to assess the efficacy of your work.

"...alternatively is there a way to tie this in with a ventilation system for the whole house to improve air quality?"

That's a good question. I suppose there is a way, BUT!
You would need to run the system full time, there will be HVAC losses unless you are in a year 'round temperate zone, and increased energy usage to run a larger motor. Sole purpose radon fans are low draw...I haven't seen one locally above 4 amp.

"Radon gas is under pressure from the strata underneath it."

Not enough to matter.

The fan does the work.

True, but the pressure is what forces the gas up and out of the ground---and into houses.

Does the vent stack have to be inside the house or could it be installed at the side of the house somehow? This is a tiny cottage room - about 500 square feet. Is this something the ordinary contractor would know about and be able to do, or do I really have to get a specialist?

"but the pressure is what forces the gas up and out of the ground"

Plain old diffusion is more than enough.

You can do the job yourself. And the pipe can be installed on the outside of a wall. I read one example where 5" ABS pipe was used to start and then connected(via off the shelf adapters) to gutter down spout material. Made the installation look much better.

Have you had testing done? Testing methods are as contentious and varied as the solutions. My opinion is that several 48 hour tests, spaced several days apart, are the most reliable method.

Re your question "have you done testing."

Well here's the situation - no, havent done that. My understanding is that testing may or may not be accurate. Meanwhile - without going into too much detail - the person whose house this is lives in an area with high prevalance of radon and has developed a serious lung problem (for which radon is a recognized cause). May be coincidental, maybe not. My thinking is if the fix is simple enough and inexpensive enough, why even go there (testing). Just get it done.

"has developed a serious lung problem"

Likely to late for them.

The only thing radon is expected to cause is lung cancer.

Radon undergoes alpha decay (throws off a helium nucleus) and becomes polonium.
The alpha is low energy and can only penetrate a few cm of air, and cannot even penetrate skin.

The problem is that if you inhale the radon and the decay occurs in you lungs there is essentially no protection for the lung tissue.