What happens to a greenhouse gas after it absorbs infrared radiation?
Actually a greenhouse gas absorbs light in the infrared wavelengths of light. One of two things will happen after the gas absorbs the IR radiation: 1) the molecules to move around and vibrate faster, which is really the definition of temperature, so the air gets warmer, and 2) the gas molecules can re-emit the IR radiation. In the second case, the radiation is emitted isotropically, so that some of it just goes out into space as it would have had the gas not been there in the first place, but some of it will be directed back towards the surface of the earth, warming the surface by absorbtion of the IR (heat).  

Note that most of the heat being absorbed by greenhouse gases is coming from the ground after the ground absorbs sunlight and warms up, not from the sun shining through the air and being absorbed by the greenhouse gases directly.