How much should I worry about painful urination?

Moderate concern. While even a "mild" urinary tract infection can have severe symptoms, there are many potential causes of infection, including usual and unusual bacteria, yeast, stds, and even other organisms. While many resolve on their own, depending upon associated medical conditions, sexual activity, age, immune status, medication use, and other factors, each case should be evaluated if it persists.
Some. This is not a symptom that necessarily means infection for which you'd need to see a doctor for antibiotics. However, if symptoms persist, you have to see your doctor please.