in cold weather before running it. Is this normal?

I was really worried this morning when I was about to drop my lil girl to school, first - i had 3x attempted to start the engine but it just goes off after turning it on then the engine managed to stay on but I can feel that it will goes off at anytime or feels the engine was weak, second when i put it on 2nd gear it doesnt kick in properly, its just very slooow. Though, on my way back home seems running OK..

Now just hoping that my car will be running OK when I go to work later

I've had my car since June this year and its a Vauxhall Corsa GLS year registration was 1993 and its done only 52000 and also I hold a full driving licence alsmot a year now, so i dont have much experienced, so hope you guys can share your knowledge.

Thanks