I am having having palpitations pretty frequently (many times a day...at rest and active) which then cause me to cough. ?

Cardiac arrhythmia : This sounds very much like a cardiac arrhythmia. Also known as an abnormal heartbeat. You need to see your doctor today and have an evaluation including an ekg. Most arrhythmias are benign and nothing to worry about but some are not so and need to be treated.

See your doctor soon: Palpitations can be harmless but annoying or potentially a dangerous rhythm problem. There is no way to tell these apart without an ECG taken while it is happening. If you don't experience them while at the doctor, you can wear a monitor to catch them when they do occur. Treatment can be simple lifestyle advice (less caffeine) or medication. Don't ignore this symptom.