General Advice From A Medical Doctor
After the session, the best treatment is simply that which one would give a bad sunburn. For the first few hours (or overnight) leave on the protective dressing supplied.
After removing that dressing, clean gently with soap and water and apply a thin film of antibiotic ointment (Bacitracin or any of the triple antibiotic ointments available over the counter). That can be repeated during the day for comfort and sticky clothing. The process is repeated the second and third days.
After that, keeing the area clean on a daily basis and (if you wish) lightly lubricated with some vaseline to minimize crusting also helps minimize the itch. Plastic surgeons keep wounds clean and moist for least scar formation. Do not go swimming for the first day or two after tattooing.