I have changed up my style to the "Goth" side, and I love it. But is it weird that I am 25 and doing this? When I say "Goth," I mean corsets, black dresses, jeans, dark makeup, etc. Nothing crazy like rainbow hair or such.
Rainbow hair is the best aspect of goth, so go for it. Aside from that, your age isn't a problem for the look. A lot of people older than you have the black clothing and heavy makeup features, no one thinks it's 'goth' at your age.