I recently started tanning a week ago and a few small red dots have appeared on my breast. they don't hurt or itch and are not raised. is this common?

Red bumps: If the red bumps seem to be around hair follicles it may be folliculitis or inflammation around the hairs. Tanning beds are a good place for bacteria from others if they are not disinfected. Could be allergy to type of lotions used. If they worsen or it doesn't resolve in a few days I'd have it checked. Hydrocortisone or Benadryl (diphenhydramine) cream may help if it begins itching.