Firstly - relax. Serious equipment failure is rare, as in something that could be immediately life-threatening due to catastrophic failure, but minor problems such as broken mask straps do happen, and they are relatively easy to deal with... but, if a diver is not capable of dealing with basic problems underwater, then panic sets in, and then really bad things can happen.
Learn to control your breathing underwater - rapid/shallow breathing will eventually lead to a diver feeling out of breath and the normal human instinct is to react with a fear of drowning. Accept that this is a normal reaction, but with a bit of practice, this can be overcome, that's why we lug all that gear around with us.
With the mask thing - it's quite normal for people to feel a bit panicky with water over their exposed noses and you need to learn to control that.
Sit in the shallow end of the pool with either a snorkel or regulator, deep enough that you can submerge your head, but shallow enough that you can lift it out of the water when you feel nervous.
Start with the mask and snorkel/reg. Lower your face into the water and take long, calm, slow, deep breaths. Next step is to remove the mask. Pinch your nose closed and again lower your face into the water. Take the same long slow breaths. Once you have established a breathing rythmn, unpinch one nostril, keep the other one closed. Try to maintain the same steady rythmn, and if you start to feel panicky, close your nose again. Repeat the process. Then repeat it with the other side of your nose, and finally, when you feel ready, remove your hands completely so your nose is exposed. Try to keep your face in the water as long as you can.
Once you're comfortable, submerge your whole head, and progress from there. I've done this with several students and it works. Ultimately you want to be able to take your mask off without closing your nose, of course, but this can be achieved in slow steps rather than just taking the plunge, as it were.
Good luck - hope it works out!
C.
