2000 Saturn SL1 2000 Saturn Cold Weather Starting problems

Every since the winter came around my car has had some trouble starting. A little over a month I left my dome light on and killed my car. After getting a jump I took the battery to get recharged after which it still had problems when cold. Also to note, once the car does starts and I drive around a little bit, it will start right away usually. Also when I turn the ignition, it turns over and tries to start the car but usually doesn't for about 10-20 seconds of me holding the ignition. However, with these problems it still starts every time. I did notice this morning that my speedometer wouldn't light up when my lights were on while I was driving. I just know it's not good to let it turn over like that. I recently had my oil changed and got the battery/alternator/starter checked at Autozone and found no problems. The guy at autozone still thinks its the battery though and wants me to let it sit for a day and drive it back up there to see if it's still fully charged like it read earlier. One final note, when I was getting my oil changed they recommended I get a flush of something but I don't remember what it was.