I do not think you need to worry
about these red spots, as they are unlikely to represent a serious medical problem.
One possibility is that you have exercise induced urticaria (hives
). This is relatively common, and occurs when certain inflammatory cells in the skin are stimulated by the physical exercise and release chemicals which lead to transient raised hives on the body. These might be preventable by taking an over the counter antihistamine before exercise, but you do not need to if they do not bother you. You would only need to worry about this problem if you also developed swelling of the lips, face, or tongue, or wheezing while exercising.
The other possibility is that you simply have some mechanical irritation of the skin from friction caused by exercise (such as clothing rubbing). You can manage this with good skin hygiene, wearing sweat wicking clothing while exercising, and keeping the skin well moisturized before and after exercise. If you have a few particular problem areas, you could apply one of those commercial anti friction products to the skin before exercising.
If the symptoms worsen or change you should talk to your primary care doctor
, but otherwise I don't think you need to worry.