What types of home radon mitigation systems have you used, and how well do they work?
Start here:  EPA's guide on radon mitigation for home buyers.

We have a radon mitigation system in our house.  Our house failed the radon test during the home inspection, so we forced the sellers to have one installed.  The work was done by a local radon contractor, paid for by the seller for $975.  We also had another bid that was $1100.  (Those prices included an independent re-test of the radon levels after the system was installed.)

There are lots of Youtube videos of radon mitigation systems.  They are worth watching.  The videos were very informative for me when I was trying to understand all of this during the home buying process.

For our system, they cut a hole in the basement slab and sunk in a 4" PVC pipe.  The PVC pipe runs up into the garage.  It runs along the wall of the garage, and into the attic above the garage.  In the attic, there is a radon fan (model RP145).  From the fan, a short length of pipe extends up through the roof.  There is a rubber weather-proofing boot around the pipe on the roof.

Our system has a u-tube monitor attached to the PVC pipe in the basement.  It measures the suction, so you can see if the fan has quit working.  Looks like this.

The fan makes some noise.  You can hear it in the garage, but it's not that loud.  Everything I read about the model of fan we have says it is good, with a pretty long average lifespan.  (They do eventually break and have to be replaced.)

The one thing that sucks is that I now have a big 4" PVC pipe along the wall in the garage.  In hindsight, it might have been possible to rip up some drywall and run the pipe inside the wall.  (Not sure.)

The radon level was at 21.9 pCi/L during the home inspection.  The re-test showed it at 0.9 pCi/L.  Here is the EPA's guide on radon levels.

For a house with lower radon levels, they may use a "passive" system with no fan.  For higher radon levels, they may use a more powerful fan and a bigger pipe.

Depending on the construction of the house, the radon contractor may want to install the system on the outside of the house (pipe and fan mounted against a rear wall, running up to the roof).  While researching mitigation systems before buying the house, I learned that an outside fan installation considerably increases the failure rates of the fan motor.

When designing/installing a radon system, some thought goes into the angles of the PVC pipe to control the influx of rain water into the pipe.  The pipe also has to be routed through the house without destroying structural studs and joists.  The wiring for the radon fan also has to be up to code (e.g. use a junction box).  A competent radon contractor will do these things, but it doesn't hurt to ask questions.

The other thing a radon contractor may do is caulk and seal any cracks or gaps in your foundation.  Unused drain holes may be closed up.  The idea is to prevent any radon gas in the surrounding ground from seeping into your basement.

If you are concerned about radon in your house, there are some low cost radon home test kits.  You can buy them at home improvement stores.  They come with a charcoal canister, which you let sit in your basement for a while, then you send it to a lab for results.

We also looked at getting one of these units and installing it down in the basement.  It does continuous monitoring, and will beep at you if the levels get too high.  Never actually got around to ordering it.