I did have what I would call a mild sore throat today and a little bit of heartburn, would you think that common in this case?

Yes. Upper respiratory infections, like the common cold, can give multiple symptoms, heartburn is one of them, especially with sore throat. If you have heartburn regularly without being sick you should speak to you doctor as there are multiple treatments for it. .
Yes, If you mean. Sore throat and heart burn together. Gargle with salt and warm water, Ginger tea will help you also.