Let’s start with this:
Movies have it wrong.
In just about every movie ever, first reaction to getting shot is to pull out the bullet, at any cost.
This is nonsense, and if you try that, it can kill you.
First aid for a bullet wound is to stop the bleeding and protect the wound. This usually involves applying a dressing and putting pressure on it. Then, get to a doctor.
Doctors will often remove the bullet, but this is because it is in the way of something they are doing, like repairing a blood vessel, or because it’s pressing on something vital. Other times, they will just leave it where it is, give you some antibiotics to prevent infections, and then you get to have interesting conversations with X-Ray technicians and, maybe, airport security for the rest of your life.
Lots of comments ask about lead poisoning.
Most of the time this is not an issue. Lead needs to be dissolved to spread through the body. Common ways to get lead poisoning are to digest or inhale lead as fine particles, and even then it takes a long time to happen. Most of the time a bullet isn’t going to dissolve. There will be some scar tissue forming around it, and it will just sit there. Exceptions to this are bullets in contact with cerebrospinal fluid or in a joint. That said, if you have a bullet in such a location, you definitely need a good surgeon to deal with it if you don’t want death or severe disability to be the end result.