Some good answers to what to do once the car is started, but are you getting it started? To me it read like you can't get it started at all.
First, you should be able to get the car running without the choke working at all. Is it easy? No. But I ran an entire winter in New England (and I drove it every day) with no choke whatsoever. Just gave it lots of gas and had to keep giving it gas for abotu 3 minutes until it would keep itself running.
Not to overstate the obvious, but without any other information, let's rule out the simple stuff:
1) What weight oil? Running too thick an oil in cold weather can really make it tough to start. my 91 explorer won't start on really cold days if I have 20w50 oil in it (dieing oil pump).
2) Are you pumping the gas to reset the choke? (this goes for manual or electric). Remember, the choke will not close itself.
3) Pull the air cleaner off, and open the choke plate up manually. Then push the throttle linkage and look down the throat of the carb as you do. Are you seeing fuel spray?
4) You may need to adjust the fast idle screw (not the idle mix screw). You may not be getting enough fuel mix to get it started. Not knowing what kind of carb you have, I don't know where to tell you to look.
Slade