Detecting Radon
A number of fairly inexpensive and accurate methods such as
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Activated Charcoal Canisters
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Alpha Track Detection Kits
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Electret-Passive Environment Radon Monitor
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Safety Siren Radon Detectors
are readily available.
The Most Cost-Effective Method 
A radon gas monitor conveniently allow short and long term readings of the radon levels. With this radon detector, the homeowner can monitor radon levels both with short-term and long-term averages. The radon gas monitor must be plugged in for 48 hours before the first reading is displayed.
A reading of more than 2 pCi/L is a clear indication that you must take action to mitigate the levels of radon within your home. In Canada and elsewhere the measurement unit may by Bq/m3. Do not get confused, 1 pCi/L = 37 Bq/m3 and a reading of more that 74 Bq/m3 calls for action.
Although radon gas is almost 8 times heavier than air, do not make the false assumption that it will just stay in your basement and your will be safe, because the HVAC system does an amazing job at pushing it to all levels of your home. Radon kills slowly and the effects can only be seen after 5 to 25 years, depending on the amount of radiation you are being exposed to. You definitely must do more than just hope for the best.