I got a new tattoo on Saturday and the skin around it is red/swollen. Is this a sign of infection?

Try putting bacitracin ointment on it. It is safe for tattoos and piercing. It is also a topical antibiotic. It cannot hurt the tattoo, or your skin if you do not have an infection therefore it is worth it. Try to avoid touching the region with unwashed hands or allowing anything to touch the area that is not clean. Skin is a harbinger of many types of bacteria, therefore the chance for infections are high if you are not careful.