My message for the OP is:-
If you have to get off and push, do it. There are some bad hills round where you are and on the bike you have, some are steeper than what the bike is geared for. No need to shag up your knees straight away.
There is no shame in pushing a 'roadrace' bike up a 14%. I've seen it done on a 200 Audax. ( Ipstones ).
There is no shame in stopping at the bottom, doing some stretches, drinking a swig of energy drink and having a piddle in the bushes.
Passers by will think you are SERIOUSLY in training. If they say anything, reply "this will be my sixth ride up here today". It will be a lie but 'who cares'?
Another tip is to start by setting 6% as your initial challenge. When done confidently, go to a 8%.
Your bike has gearing for a 22%, but DON'T try this grade until you have progressed steadily 2% at a time.
It's a bit like weightlifting, only an idiot tries to lift the whole stack on the benchpress at their first session.
Fitness and strength DON'T come instantly. Be patient and tackle each steeper gradient when you have truly conquered the grade you are at.
The 'breathing' tip was good. Breath out on the exertion, inhale on the little bit of relaxation. This means a slow cadence, in the same manner as an 'alternate leg press' on a seated squat machine.
Starting on lesser steep hills eg 6%, you can practice getting the cycling form smooth and correct. Hammering away at a steep hill with high crank revs will risk straining the knees and hips, and is not good form. The last thing you want is a hernia.
Have fun.