A roping horse is used for speed with *very* fast stops once a calf is roped. This would NOT be a good horse for a 3-year-old. Usually the horses chosen for training into this sport can be hot natured and spookey. She could be walking calmly one moment and charging around the pasture the next if she is surprised by a bird, critter, loud noise or what have you.
If you are not experienced, a horse may sound like a lot of fun, but they are a lot of work and can easily become sick or injured without just the right care. Even ponies labeled as "easy keepers" can have issues, like foundering on grass in the summer.
A horse's hooves also require care every 6 weeks... no exceptions on this rule as overgrown feet can cause injury or permanent damage. They must be wormed on a rotating cycle of different varieties of wormers.
If you're willing to take on all the responsibility a horse requires, what you want is an old pony. You want one that has seen and done it all - that won't spook at things (much) and that doesn't have any nasty habits like biting or kicking. Something older than 14 or 15 is a good bet if it is sound. I'm not a huge fan of Shetlands. Look for a Welsh cross or a Quarter Horse pony cross. Look for one that has had plenty of training and lots of riding, and has perhaps just been outgrown very recently. Horses that have been turned out and left alone for long periods of time can develop bad habits too.
If you consider yourself inexperienced, I would say, "don't buy a horse just yet." Go get some lessons, take your daughter to places that offer pony rides, and then take your time finding just the right (safe) animal.
Best wishes, and have fun on your journey learning about horses - they are wonderful if you're willing to do a lot of work and spend quite a bit of money