1 second is not 3 car lengths unless you're going really fast. Also, when pulling away, you shouldn't sit stationary and wait for the gap to establish, as by the time you're going you will be even further back. Instead, pull away at the same time as the vehicle in front, but restrain your acceleration so they still pull away from you. When they stop pulling away, you stop accelerating and you'll have more or less the correct gap.

Standard advice is usually 2 seconds in the dry, 4 in the wet - "only a fool breaks the two second rule" takes about 2 seconds to say, then "but if it pours, make it 4" brings it to 4 seconds.