What size coop would you suggest? (update)

With 15 birds, your minimum coop size would be 60 square feet, which allows 4 square feet per bird, which is minimum recommended. With a run, the minimum recommended is 10 square feet per bird, so that would be a 150 square foot run. With a coop that size, that you want to be able to walk into, and also keep under $300, I would think you would want to look into something like a hoop coop. My coop is a hoop coop, and if you click on "My Coop" under my avatar, you can see mine and how it was built. Mine uses three cattle panels, but for your amount of birds, two panels would work. They are also fairly portable and easy to move and take down, which might score points with your parents if they are worried about having a huge coop around forever after you build it.

I also want to add that if you want chickens, you are definitely going to need a job. On top of the expense of the coop itself, and then the birds, you're going to need a lot more stuff to get going. You'll need feed, which is usually at least $15 for a 50 pound bag depending on the brand and where you get it. If you're raising them from chicks, you'll also need a heat lamp, a brooder, and special chick feed which costs more than layer feed. You'll also need to buy bedding to change the coop when necessary. So unless your parents are willing to spend probably around 30 or 40 dollars a month each month after you get them, depending on how much feed and bedding your flock go through, you will definitely need a job to support them. You may be planning on egg sales to help with the cost but remember, it takes about six months for the birds to start laying. But it can be done. I'm a 21 year old college student working part time and I can manage all my birds, so don't give up hope!

 

FACT: Most coop builders grossly over claim capacity... More room is generally always better, so if they say it can house 10 chickens assume it can realistically house 3-4 comfortably... Also consider that coop size is only one factor, with a big run or free range the coop does not necessary have to be as large to comfortably house them...

There are are lot of DIY cheaper options, either built from scratch or re-purposing a shed or what not...

Also, unless you get a lot of free stuff or dirt cheap 2nd hand stuff, $200-$300 isn't going to go far in building a coop and run... Sorry to be blunt about that but I now how fast you can drop $100s on just the basic building supplies like hardware wire for a run... That isn't to say if you are resourceful you can't do it on the cheap, lots of people do wonders with free stuff like skids and stuff they found laying around...

I would personally suggest you keep an eye out for used sheds in your area if you want to save a buck initially, they provide a cheap and easy way to get a basic structure to build upon...

 

So what are your weather extremes? Cold? Snow? Heat over 95? Humid or arid?
Is the property fenced? These all play into your coop design.

If your folks like to keep the yard nice close by the house, you do not want the coop there. I had 3 chickens (2 now) that destroy any plant they like, move dirt and mulch like bulldozers, and leave holes everywhere they go. I 'used to' have a nice yard! They will do okay at the edge of the wooded area.

Do you have a plan for what to do with all those extra eggs?

Personally, I think 15 is a lot to start out with. If you have fewer, you will get to know them better. If you have a problem, you probably won't lose as many in one fell swoop.

Pick your two favorite breeds, go the breed threads here, and get recommendations for where to get high quality birds with good personalities.

Your enthusiasm is wonderful, but if you get a job, your parents will be left with responsibility for birds they did not want in such a large quantity, no? You can always expand later.

 

Pyxis gave you good advice on the size of a needed coop.
Here are some thoughts I had. Start with the biggest coop you can. You're always going to want more chickens. Chicken math, you know.
But on the number of chickens, you can start with a smaller number and add more as you feel the need. And decide which breeds are your favorite. You mentioned 15 being the minimum number for shipping.
Some suggestions. Depending on where you live My Pet Chicken will ship as few as three chicks. Where I live four is needed, but I never got that few.
Perhaps you could combine your order with another person or two to make the minimum shipment. Sometimes local feed stores and hatcheries will sell less if you pick up.
Did you want birds for meat or eggs? Unless you want extra eggs to sell, six laying hens should provide enough eggs for a family. When I first started in chickens four hens provided enough eggs for our family. (But that didn't stop me from getting more.)

 

I have a hoop coop up here in Vermont, and it works very well during the winter. It's really strong too and can hold a ton of snow. In the last big storm, we had three huge tree branches come down on it along with all the snow, and there was no damage whatsoever. It didn't even bow in a tiny bit. So yes, they do hold up well to winter