Dairy yes, gluten only applies if you have an intolerance or sensitivity. If your body digests gluten without a problem, there's no reason to cut it out. Most gluten free substitutes contain more fat, sugar and/or salt and are usually far less healthy and would likely be worse for your skin too.
Dairy does affect some people's skin but again, doesn't mean it's bad for everyone. But might be worth experimenting with.
Generally good healthy and hydrated body is the best you can do for good skin health. So make sure to eat a balanced diet, drink lots of water and avoid indulging things.
Other things to consider would be stress levels, sleep patterns, exercise etc. anything that can affect your hormones.