welcome! I don't mean to put a damper on things, but I think a 10 year old is way too young to have a horse.....
I got a horse when I was 10 (almost 11) and was from a completely un-horsey family, and I had been taking lessons for about 2 and a half years. I don't think I was too young, I had a lot to learn but I had read a lot of books and was really eager.
Be sure its what you want though. Its definitely a decision that your parents should be very involved in, maybe get them to come on here and look around at the recommendations. I'm sure you're pretty smart and capable
but sometimes when you go looking at horses you like them sooo much that its hard to make the best decision. Remember one of your parents is probably going to have to go out to the horse with you everyday, so make sure they know what they are getting into as well.
I would look for an experienced horse around 14 - 15 hands. If you are going to be buying your horse be aware that you might want to sell it in a year or two if you decide you want to get into showjumping or western or dressage, or if you grow really tall. In my opinion you don't want to buy a really old horse unless you think you can keep it forever, so instead maybe look for a 15 year old who has taught a few kids in their time.
Leasing a horse is ideal for a first horse/pony if you can find a suitable horse near where you live. My parents were only going to lease first, and a horse that we were offered after riding was an 8 year old 16.3hh OTTB. What kind of horse owner offers that to a ten year old! Luckily horse people had told me my mother not to buy a TB so she turned him down. If you lease a horse its okay to get one that looks a bit old, maybe in its 20s, or a pony that you are going to grow out in a year or so, just look for something nice and calm and as safe as possible.
Breed wise I don't think it matters much, but probably keep away from Thoroughbreds are they are generally less forgiving and more unpredictable. A nice Quarter horse or QH cross might suit you really well, as they generally have kind dispositions.
When you look at a horse take your parents, your riding instructor if you can and also get a vet out just to make sure the horse is healthy. It costs a bit more money but its definitely worth it.