First of all, sorry if this doesn't go here - I haven't used this very much.
I have been riding for 6 years, but have never owned a horse. I am an advanced rider, I am into hunters/jumpers, and I have ridden many different breeds of horses, from QHs to Morabs. I want a breed that has some 'go' in it, but is easy to handle and won't bolt on me. I am also into breeds that are able to jump up to 3 feet or higher, as I currently am jumping around 2 feet. The breed must be not too tiny for a 150 lb person, but not to tall for kids/teens. It should usually have a good temperament, meaning around a 2 or 3, no more then a 5, if you rated it 1 the calmest, 10 the most temperamental. What are some breeds that meet these specifications? Also would want a horse that could be used for lessons if needed.
Any, but the hot breeds or very large draft breeds will produce animals that meet those requirements as will a great many crosses. Most breeds will produce animals that are in the 15-16 hand range. 16 hands might be a bit for some kids although my 8 year old granddaughter has ridden my 16.1 mare.....she just can't mount from the ground without a leg up
I'd advise looking around and checking out a variety of horses that aren't hot or large. It's really going to be mostly about the horse than it is the breed. A breed can produce one horse that will jump over 3' with no training and another horse that won't want to jump 20". One horse that will stand it's ground with the first barking dog it encounters and other that will spook the first time it sees a chicken. A horse that when spooked will run 30 feet before it stops spooks and another that will run a 1/4 mile before it slows down.
For no more than what you're looking for (which isn't a lot really) price will likely drive what you get more than breed.
To narrow the list I'll go back to what I started with. Skip the hot breeds (Arabian, TB, etc....) and the very large draughts (Belgium, Shire, etc...). Not that you can't find a hot breed horse that's not "hot" or a draught breed horse that's light enough, but the odds are lower.