Yes there are a couple of things that could go wrong but, most of them can be avoided if the tattoo is done  by a professional. The biggest point of concern of many is the actual needles used by the tattoo gun. They can, if reused, cause blood carried diseases like STDs. To prevent them, be sure you ask the tattoo master to see him open a new package of needles place them in the tattoo gun before touching you with it. That's what I did to calm my nerves before making my first one.

Most other problems associated with getting a tattoo come to be either by lack of skill by the tattoo master, or lack of care by the person who got the tattoo but some are generally present and should be considered before getting a tattoo done. To prevent them, do your research and find a good, licenced professional to make your tattoo and ask him/her how to propperly take care of your tattoo when it's finished. 

The risks of getting a tattoo include:

• You might be alergic to the ink
• If a metal based ink is used, when going for an MRI exam , one may get  MRI-induced burns due to iron compounds in their tattoos   
• It could hide skin cancer
• It could get infected
• There is a risk of getting an STD, or other blood borned diseases
• It could affect how you sweat -> Studies have shown that tattooed skin sweats 50% less than regular 
• If not cared for propperly it can cause scarred tissue

The top tips for caring about a fresh tattoo my guy gave me were to:

• At all times for the next week keep a thin petroleum jelly and a bandage over the tattooed area.
• Gently wash the tattoo a couple of times a day with cold water and an anti-microbal soap. Pat it down to remove any water, apply a thin layer of petroleum jelly and place a bandage over it. 
• Always keep the tattooed area "wet" with the petroleum jelly and never let it dry out because it can lead to scarring and loss of pigment.
• After a week you can stop using a bandage to cover your tattoo and leave it uncovered but still apply a layer of petroleum jelly to keep it "wet" at all times
• For the first two years after getting it its suggested to,  if  exposing the tattoo to sun, to use a strong sun protecting creams (factor 50+)

To conclude

There are some pretty scary side effects that can happen when getting a tattoo but, most of them can be avoided by chosing a professional tattoo artist and good care of the tattooed area. I have a couple done and never had any issues whatsoever.