2003 Monte Carlo wont start

Hello, I have a 2003 Chevy Monte Carlo LS 3.4L with 51,000 miles. Help! My car won't start when its cold! I tried starting it when its cold and all I get is a battery light that comes on. But, all my lights come on and so does the radio! So I got a new battery recently, which an auto mechanic told me to do so after the computer read ''bad battery'', but I tried starting it after that and now the car will start, but it'll cut off quick! The only way I can get it to run is if I push on the gas while im starting it! And this only happens when its cold outside! And by the way, I just recently had a new fuel pump put in! Help! Im tired of spending hundreds of dollars for nothing!