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Any sign of infection take them out, let them heal and re piercer them at a later date. It is worse case scenario, but the infection can travel into the body and for someone so little it isn’t worth the risk! But it shouldn’t be an issue if kept clean and well looked after which I have no doubt you will do. Don’t change until they are fully healed, so wait until 6 weeks minimum. They might be a little grumpy for a day or two as they will be sore, even as an adult piercings suck, so calpol may be needed for the first night sleeping as they will be sore when knocked and laid on. I’m sure everything will be fine though, I had my ears pierced as a babies, just like thousands of others and we are all clearly fine. Just expect a little grumpy baby.
maybe wait until they can consent to the piercings themselves, most reputable piercers won’t pierce children under 10.
As a professional piercer I wouldn’t personally pierce anyone’s ears under the age of 11, it isn’t something I agree with, but of course it is your child and your choice, and I respect that. If it’s something you are going to go ahead with, would recommend giving them calpol after it’s done, as the shock is still going to scare them and make them cry. So as soon after would be best as they will be sore and they will be throbbing. Be warned that you may only get one done, as the first one shocks them and makes them cry, leaving them having a meltdown and making it extremely hard for the piercer to then do the other one. It might however be a little easier at 3 months old as you can hold them still a bit easier than a toddler. Just be careful and make sure to keep them clean. You can get a saline solution for piercing aftercare, or simple use salt water. Clean them 2-3 times a day, and at any sign of infection, just take them out. They can always get re-pierced.