If your tattoo is located on your inner arm (biceps) you'll want to avoid running or any exercise that causes your arm to brush against the side of your body.
If your new tattoo is on your lats there's no reason why you can't train legs at a low to moderate intensity a few days later (provided you don't sweat excessively).
If your new tattoo is on your ankle or the back of your calf muscle there's no reason why you can't hit a shoulder workout a few days after getting your fresh ink (once again, provided the intensity/sweat isn't too high).
As I'm sure you've concluded from the examples above this comes down to common sense, don't train a muscle group too close to that of your freshly inked area.
The Intensity Of Your Training Regime
Excessive sweat is far from great for a new tattoo during the healing process, although there's not a great deal of research out there as of yet as to how much sweat can actually affect the healing process of a tattoo it's safe to say it's definitely not beneficial. Therefore light cardio and low intensity weights/bodyweight workouts are going to be back on the cards long before HIIT sessions and 1 rep max training.
The Hygiene Of Your Training Locations
The gym is undoubtedly one of the biggest breeding grounds of bacteria you'll find - from the sweat left by other gym-goers on machines to the amount of dumbbells/barbells/machines that gym-goers have touched without washing or sanitizing their hands... the gym can be a dangerous place when you've got a fresh tattoo (which is essentially an open wound) when your gym's hygiene isn't tip top.
If you train in your own private gym or a very small and well maintained boutique gym you're going to be able to resume training sooner than someone that trains in a poorly maintained 24 hour chain gym.
A Few Tips For Your Workout After Getting a Tattoo
Wash Your Hands Immediately After Training
After you've touched any equipment, be it barbells and dumbbells or even a basketball you need to wash your hands immediately as if you touch your tattoo (even by accident) after handling fitness equipment that who knows how many people have touched you've just increased your risk of infection big time.
Don't Train In Anything Too Tight (Compression Wear)
Constant rubbing of your new tattoo against tight clothes such as compression shirts and shorts as well as thermals are a sure-fire way to irritate your tattoo.
Not only is this super uncomfortable you'll find it'll quickly get itchy.
I mentioned above that the gym is a breeding ground for bacteria, well swimming pools, lakes and water holes are completely next level when it comes to the risk of infection.
Body oils, saliva, urine, sweat... swimmers leave it all behind in the water. Even if the pool you're swimming in is correctly treated the chemicals used to do so can cause terrible irritation too - avoid swimming at all costs for as long as you can while your new piece of artwork heals.
Besides the risk of infection from bacteria your ink is susceptible to fade away and breakdown due to being over-moisturized.
Limit Your Outdoor Training (Sun Exposure)
If you opt to train outdoors ensure your new tattoo is covered (but not by anything too tight as mentioned above).
Excessive heat and sunlight exposure often causes itching and peeling.