Hmm, how do I like her? I LOVE HER!!!

Howdy!  My 81 does the exact same thing even after I had the carb blueprinted.  Its a q-jet thing, no way around it. 

WHen I start, I give 2 pedals to the floor, then turn the key and go about 1/2 way down maybe a little more.  It'll take a few secs..  My 81, I wanted this since an old girlfriends dad had 1 when I was in college back in 81.  I love it! 

Just to give you some idea how your choke works. Gas engines are built to run using a certain fuel-to-air ratio when they are at operating temperature. That's when your carberator, in this case, would be adjusted for peek operation. When you turn over any cold engine it takes a higher fuel-to-air ratio to start the engine. The choke has a flap inside the carberator that closes off most of the air entering the carb during cold start-ups, thus "choking" the carb. That flap is attached to a small rod that attaches to a small flat metal coil that sits no top of your manifold just below and outside of your carb, and as the temperature rises from the heat of your engine that coil expands, turning the rod it's attached to and slowly opens the flap allowing more air to be mixed with the fuel. When you shut your engine off and let it cool, the coil then contracts and should close that flap again. BUT, the flap doesn't close right away. Just before you start the cold engine next time, you need to press your gas pedal to the floor once. This will release the flap to close and squirt a shot of gas into the carb. And now with the carb choked and that extra gas in it you should be able to start the engine. Depending on how much choke you have will determine how quickly your engine will start. Sometimes you may have to press the gas pedal a couple of times to get even more gas in there before it'll start.

P.S. For easier starts in cold weather you should use a light weight oil. 10W-30 is good.

With today's cars being fuel injected, we get used to that immediate start up. It's just not so with our older Corvettes. They're not (most of 'em) controlled by computers, they're mechanical instead.

I would like to know if there is any smoke on that first start...  if it is black then you probably have fuel leaking down as joel suggested.  if it is blue/gray then you could have oil leaking past the valve seals.