I belong to a concierge practice and can normally get a same day appointment to see my primary care doc. If it's something simple, that takes care of it. If it's more involved, I'll get a referral to the best specialist she knows. If it's really urgent, she wants me to go to an urgent care clinic. Around here, an urgent care clinic is usually called a "doc in a box" and they can handle quite a number of problems on the spot. If it's tougher than they want to deal with, they just tell you to go to the nearest hospital ER.
The concierge practice is expensive, but I feel it's worth it. It's also part of a nationwide system, so if I'm traveling I can call my own doc who will tell me of the nearest associated practice to where I am, and they will see me right away as a courtesy.