My son developed a fever, cough, runny nose, and watery eyes. Most of those symptoms have cleared up except the runny nose and cough. Now he has a rash that started as reds spots on his neck then became bumps which has spread down his back and on his face.

By taking him to his doctor

Measles isn't the only concern.

The only way to get a diagnosis is to see a doctor. There's no way for anyone to diagnose on the internet. There are simply too many possibilities for the symptoms you described.