Hello and Welcome!!
First off, in my opinion, there are few things that can shape a child into a good citizen by teaching them to care and raise for animals properly. Ask yourself if you are truly given to the responsibility of caring for these animals, day and night, and to buy the food. Once you are truly sure you are committed to this, you can approach your parents and ask for their permission. Remind them that responsibility is a big thing for a person to take on, and that it will be a character building experience for you. Don't forget to remind them of fresh eggs and the positive environmental impact! (no more buying from factory farms)
A chicken coop can be free or expensive, depending on where you live. If you live in a warm climate with few predators, you can go with simple mesh fencing, posts, and a doghouse (igloo style!) In a cold climate with predators, you will need to fortify against ice, wind, foxes, et. We'd have to know what type of hazards to deal with to point out what you can save $$ on. My perches are free, throwaway lumber pulled off the curb garbage pile and nailed together (imagine an H with two cross bars..)
We live near a college town, where, every year, college students throw away thousands of dollars worth of lumber in the form of loft beds. We have kept track of dorm move out days and saved literally hundreds of dollars of lumber. Also, have you tried posting on craigslist.org for people that want wood hauled away?
We live in a country of waste, and it is AMAZING what you can get from people that simply toss stuff away. Dog kennels, fencing, etc.
And NEVER underestimate the power of bartering labor with a farmer/mother. I'd gladly help build a coop for a responsible person to help babysit/clean house!!
Hope this helps- post more specific questions about your needs for your area, what you have available, and I'll do my best to give you some more specific answers!