QHs are IMHO the best all-round breed, but you can find a decent all-rounder in many breeds. It just depends on the individual horse.
I would recommend finding a trainer that does multiple disciplines and ask what they look for in their horses--at my barn we do h/j, eventing, dressage, cutting, sorting, penning, etc.
I like short-backed horses with good conformation, dense bone, and enough athleticism to jump, cut cattle (or have good lateral movement for reining), and collect comfortably. Generally 14.2-15.2 HH, although bigger may not be a problem since you're not interested in working cattle. If you're wanting to compete western pleasure, you'll also need a horse with a quiet mind that won't mind the slow pace. Breed is secondary to all that, haha.
My trainer has the best little QH that I got a two 1sts in a hunter division, and I took him the following weekend to a ranch sorting where he did well, then a couple of weeks after that, he got grand champ in jumpers (with a young rider). If my horse is out of commission, I sometimes take dressage lessons on him (though my trainer will admit that it's because she likes me to school him after the kids ride him, bahaha). He'll carry any kid safely, but he's well trained enough that I enjoy riding him too at a much higher level. He's really a horse that does anything you ask him to!