All the information you need for getting accredited as a coach is available on the Gymnastics Australia website. You need to do your coaches course.
The first step is to complete the General principles course, these days this is done online, you can find it on the Gym Australia website. Once you have done this you need to complete a 1 day discipline specific course, which for you I am assuming will be WAG. Level 1 WAG courses are usually held in January in all states so that will be worth doing.
After the course you need to complete the booklets that you were given at the course which are all about what you will teach in class and then complete your assignments. You assignment will be to design a term plan and then take 3 lessons from that term plan and design specific lesson plans for those lessons.
Then you must teach these lessons and have a supervisor observe you and fill out the booklet. Also you must complete 40 hours of supervised coaching.
Once you have done your courses, completed you booklets, completed you assignments, taught those lessons and had your 40 hours of coaching signed off on you send it all in to be evaluated.
If it is good enough you pass, if not they will send it back and you will need to fix up those parts that are not acceptable.
You have to complete all this work within 3 months of doing your course, if you don't then you need to apply for an extension, if you don't have an extension then you have to do the whole course again.
Once you have passed your level 1 coaches course you can't coach yet, you must apply for registration. You have to re register each year to keep your coaches certificate current. To do this you pay a fee and have to complete at least 6 updating points per year (except in your first year), you get points from attending various course throughout the year.
To do all this you must be at least 16 years old.
Almost any gym will take you on as an assistant coach while you are doing the course, they know you must complete your 40 hours supervised and we all want more accredited coaches around so I am sure your gym would support you and take you on. You would not be teaching classes yourself but you would be assisting with a more experienced coach.
However, while you are completing your course it is unlikely that you will get paid, mostly these hours are done on a voluntary basis. This is because the gym still need to employ a regular coach to teach the class and supervise you.
Its fine to coach and train at the same gym, this is how most young coaches get started, they become assistant coaches while they are still training and competing.
You won't be paid a lot to coach at first because you are very young and inexperienced. I don't know what minimum wage is for someone your age but I think its about $10 per hour. This amount will of course increase as you get older and more experienced, teenagers get paid very little.
Many gyms don't pay their teenager employee's in cash either, for many it will be fee relief instead. So you get a certain amount of money off your training fees or even free training if you coach enough.