Code:
awk '{
if($0 ~ /\[first\]/){output="off"; next}
if($0 ~ /\[second\]/){output="on"; print; next}
if(output == "on"){print}
}' input-file
The above is an untested, but rather simple, way of (I believe) printing what you wish. If it doesn't work, and/or if you would like explanation of it, please explain or ask more.
There are myriad ways of accomplishing this so I look forward to seeing some other methods too.
Good luck!
EDIT: See the bold part - I fixed it.