2007 Mitsubishi Galant Possible Alternator Issues

Drove my 07 Galant Ralliart downtown with no issues. Returned home and approximatley an hour later went to leave for work. Vehcile wouldnt start. Jump started the car and drove with no problems. Stopped for about ten minutes, then continued to drive at which point i was cruising at about 35mph when the rpm would accelerate/jump a little and go backto normal. Stopped the car andshutit off to eat. Had to jumpstart again. After running for approximatley one hour while driving home the headlights shut off with no warning. Not sure where to start but i am thinking with the battery and maybe the alternator.