Have had a cold for a week with no signs of letting up, also having breathing trouble, what is it?

See a doctor! Chest colds last longer then other colds and 7-10 days of symptoms are not unusual. There should be a sense of getting better. Pneumonia (bacterial infection in the lung) or asthma (tightening of the lung tubes) are both possibilities. Any time you have breathing trouble, no matter how long, is a reason to see your doctor asap!
Get help. The worsening symptoms this far out could be a pneumonia, sinus infection or asthma. All of these things can be easily treated if takec care of quickly and are potentially serious if not treated, so see your doctor quickly.