Help Me Diagnose... Exhaust Problems?

[LV2DRFT]

Hi, all-

First post

So I have a 2003 Solara with the smaller engine. Temperature where I live gets down to 45-50 degrees in the AM, not too bad. I'm experiencing the following symptoms:

1) Every time I start the car up first thing in the morning, I notice a decent smoke cloud come out of the exhaust. It smells kind of "fuely" but not extremely so. This lasts about 20 seconds or so, then seems to be fine afterwords.

2) Sometimes the engine will cut out completely after I stop at a stop sign, then punch the throttle down. In these cases it will do it immediately, as in, before the car starts moving forward after I step on the gas. Also sometimes the car will start to choke up a little bit when idling and give me a little bit of vibration.

Anybody know what this could be? Please help me diagnose so I can fix this myself and not get ripped off by the dealership

Thanks in advance!