Hello UG,
I'm a 17 yo guy and I've recently started taking Vocal Lessons.
Now they have been fine but there is one thing that I am not 100% sure about yet.
The only real concern I have is, when breathing should I inhale through my nose or through my mouth? My teacher says I should do it through my nose, but I find it kinda hard filling up my diaphragm fast enough to sing certain phrases. And if breathing through my nose is the right way, do you guys have any tips on how to do it faster?
I don't see any problem doing it through the mouth. Some people have problems with it because it's "noisy".

Breathing is a matter of practicing, ask your teacher for good breath exercises and just practice it. Breathing through the nose is ideal since it will remove the breathing sound in between phrases but it's not that practical.
The answer is, both. Breathing in through the nose is more of an exercise in relaxation when warming up, and can help some people 'get into' their lesson, if that makes sense. I often breath in through my nose, mainly because I'm a Musical Theatre performer, and there will be some situations where breathing in through my mouth just looks plain weird, as it's unnatural, however, I will always breath in with my mouth in a song, as I need to take in air fast and completely fill myself up with air, and breathing in through the mouth prepares you for the next note, so it can look incredibly natural, on the other hand.

As for the whole 'breathing in through the mouth is noisy thing', it's not, it's just whoever complains about that is rubbish at it. Breath in through your mouth without the noise, bring air through your mouth and fill your diaphragm, don't raise your chest like you're shocked, just keep it relaxed and focus on not making a sound. What goes in comes out, if you breath in on an exasperated sigh, you're gonna make a pretty stressful, sometimes quite breathy sound, so avoid noise when you breath in.