I am glad you are asking the question. Kids and ponies is a bit of an issue of mine as there were a LOT of kids on my old yard - some really conscientious and others who had no idea what they were doing and their horses and ponies really did suffer. Not through unkindness or lack of care/effort but just through ignorance and unhorsey parenrs not helping.
There are 3 elements to focus on: Money, Time, Experienced help
1) Money
The cheapest part of a pony can be the initial purchase price. I have known kids be bought a cheap pony who they can't afford to look after properly. They figured out a base-line livery/forage/bedding price which they could afford but there were ALWAYS extras which they couldn't. Fine if mum and dad step in, Not so fine if not. I have known a horse losing weight who needed hard feed the child could not afford, a pony who was lame but the child could not afford the vet and a strangles outbreak that required all the horses to be blood tested and sometimes scoped which a child could not manage either. In one instance a pony was removed from a child's care. Other times, liveries or YO stepped in but you cannot expect that. You need a contingency fund for vet, dentist, farrier/trimmer, box rest (more bedding), feed, breakages etc etc etc
So you either need a part-time job to pay for what the pony might need or parents who understand the costs and will support you. It is not fair on horses to have them and not be able to pay for their care. And once a pony is lame or ill, you can't sell on either. It's not really fair on the child either as it's an awful positon to be in. So if you don't have money coming in yourself, then make sure your parents are properly on board with it.
2) Time
Not just time in summer at grass, but in winter or on box rest. What are the facilities like? Can you get any help with turnout/field checks/bring in? Again some kids I know walked up before and after school come rain or shine all year round. And others left horses in filthy boxes till 4 because they ran out of time before needing to get to school. Or messaged the yard every single day for help. People do help each other but not every day. Other liveries get annoyed and stop offering to do the same pony day after day for free. This goes back to money too - if you lack time can you afford to pay for assistance?
3) Experienced help
For any persons first horse, they will need experienced help on hand. Can you tell if a horse is lame? Do you know when a vet is needed if a horse is a little off colour? What can wait, what needs a vet in the next few days and what is an absolute emergency? Whether your tack fits (more or less)? When a horse is too fat or too thin? What to feed and how to adjust for various issues? Can you put on a poultice? Or treat minor wounds? Or deal with mud fever? Probably not. Nor could I! But until you learn those things you really need someone who you can ask. I was horrifed at my first chipmunk horse and would have called the vet if my YO had not been able to tell me they were harmless grass glands.
So think it all through very carefully. And if all the issues can be managed - enjoy pony shopping and good luck!