There are a number of different causes for skin rashes but, fortunately, few of them are medically serious. The physicians who will be well qualified to discuss this issue with you include your primary care physician
or your dermatologist
As you have suspected, the rash you have is mostly likely related to exercise. Rashes are commonly associated with vigorous exercises, and the most likely cause is simply irritation of the skin by chafing against the running clothes. Rarely an allergy to some component of the fabric or to a cleaning product can produce a similar effect.
Your fitness store specialist can likely introduce you to several product lines of skin barrier creams and pastes that are designed to protect sensitive areas of the skin during running and to minimize chafing. Similarly, changing the type of fabric of your exercise clothes can at times be helpful. This is not so much due to a problem of allergy as to the fact that cotton and some other fibers are non-wicking and can leave the skin moist and susceptible to irritation.
As always, the diagnosis and treatment of your specific condition will require a physical examination by your personal physician. If the symptoms persist, scheduling an office visit with your primary care doctor
may be advisable.