Horse prices depend on a lot of things, including how much the seller thinks of their horse. I recall seeing several poorly put together draft crosses that were registered as American Warmblood, either green broke or not even started, being sold for $10k+ because the sellers thought they could charge that much just because the horse was a "warmblood." You'll see the same thing with any breed when the seller thinks their horse has 'amazing' bloodlines or has a half sibling that's a world champion.
For a first horse, I would advise against going to an auction, and if you look at rescues, ensure that they are reputable ones. Some are, unfortunately, just horse traders who are looking to make a profit even if it means being somewhat less than honest about the horse.
Do you/your sons take lessons anywhere? Trainers are great resources for horse shopping, both in that they usually know who is selling (especially horses that may not be actually listed anywhere), and that they can help you select a horse that is suitable to your skill level and desires.
In my area, you can usually find a reasonable trail horse for $1-2k. Anything under that price range tends to have behavioral/training issues to work through or some kind of physical ailment- not good for a first time horse owner and not good for a kids' riding horse.
“The horse is a mirror to your soul. Sometimes you might not like what you see. Sometimes you will.” - Buck Brannaman
"Nothing forced can ever be beautiful." - Xenophon