There's a presumption that 'nothing' can be done once the pain starts to manifest. And yet, that approach couldn't be further from the truth. Tackling your injuries head-on and/or addressing the pain as soon as it starts is the best approach to healing quickly.
Ignoring minor aches and pains can only be done for so long. Most people are attune to their bodies. Often times, we are able to determine if the injury is to the muscle, bone or connective tissue. And of course, if you're not, make your way immediately to a doctor or sports medicine specialist.
As long as you're not dealing with a break or sprain, beginning to R.I.C.E.—rest, ice, compression and elevation—from the beginning will help to keep you mobile and healing. This treatment, for soft tissue injuries, should speed up your recovery as well as minimize the discomfort you feel.
Keep in mind that R.I.C.E. is considered a first-aid treatment, not a cure, with objective of managing internal bleeding and inflammation.
Rest is a key component to heal the body. If your routine is to run each and every day, you'll need to change your habit. You don't have to stop all running. But instead of running every day, why not take the pressure off and cycle instead?