Bedbug bites can be frustrating and it is possible to transfer bugs from one person/object to another. You should discuss your symptoms with your physician and allow him/her to see the red bumps in person to get a better sense as to what might be going on and to ensure that you are not having a severe reaction.
Bedbugs are arthroprods and part of the family of Cimicidae. They are relatively small, measuring less than 1 cm in length, and are usually red-brown in color, though can become a more violet color after feeding. They can be found in various objects including furniture, floorboards, and small cluttered areas. They often emerge at night to find food and are usually attracted to heat, sweat, and odor. Once bedbugs bite their food source, they inject a blood thinner and numbing liquid into the skin. Because of this numbing component, the bites are usually painless, but an allergic reaction can develop, leading to significant itching. Often, the bites are small red lesions, but can become larger if an allergic reaction develops or can become infected and develop into a cellulitis.
If you think you have an infestation, you can speak with your physician regarding possible nest sources and eradication techniques. Regardless, you should see your physician to ensure that the bumps on your skin are benign bug bites and not from another pathology.