Cold Start problem

You didn't mention what car you have. I am assuming a carb 308.
Have someone stand back and look down the carb throats. This involves removing the airbox cover. You sit in the car and push the gas pedal down. Have them tell you if gas squirts in each throat of the carbs. I have a had my carb 308 for over 6 years and 114K miles. They start up great cold once tuned and set correctly. I wouldn't pull the choke. It isn't neccessary.

If it runs ok after then you probably don't need to do this but I would remove the idle jets clean them. It doesn't take long.

 

That is key. The fuel pump should run on a carb car when you turn the key to the first position. Then you let the pump run for a few seconds and fill the carbs. Then crank.

Check the fuse to the fuel pump. Make sure the contacts are clean and the fuse isn't blown.

There is a pressure switch in the driver's seat that most people disconnect that shuts off the fuel pump if you are not in the seat. The car will then die a minute or two later when the carb float bowls dry up.

If it starts when you dump fuel down the carb throats but won't start after pumping the gas, you should rebuild your carbs. Pumping the gas pedal should squirt fuel into the throats of the carbs via the accelerator pumps. (It is doing the same thing as manually dumping fuel in there.) If the accelerator pumps aren't working, the diaphragms on the accelerator pumps need to be replaced.

Birdman