I definitely wouldn't recommend a mini for riding purposes. They will outgrow them before you know it, and unless your kids are super young they are likely already too big. Go with something a little bigger (13 - 14 hands) and pleeeaaase don't get something super young. Ponies tend to live very long and useful lives and young, untrained horses are a terrible idea for young/inexperienced riders. Something in the 10 - 14 y.o. range with lots of experience on trails, at shows, and in the arena is ideal. The market is bad right now so you can get a REALLY well trained pony for a very good price, you just have to look and be picky. Also, I would highly recommend taking an experienced trainer with you, or at the very least a very experienced friend. They should be able to pick apart the good and bad when it comes to conformation and temperament and tell you if the pony is a good fit for your family. Remember that your kids will inevitably outgrow a pony or may lose interest, so purchase carefully and find something that other people will want to buy too (in other words quiet, well trained, sound, and with good conformation).
I would highly recommend joining Pony Club (or 4H if your daughter rides western). When it comes to horses your kids will learn way more about safety and horse care in pony club than 4H - I did both and PC is much more focused on education and creating good horsemen/women than 4H, which was more focused on showing (at least in my area). Both can be great fun and give your kids a wealth of experience.
As far as housing... Shelter will depend on your area. In the southwest we had pipe corrals with open sided shed for shade. We didn't really have grass turnout. Up here in the PNW it is WET so you really need some kind of shelter, IMO three sided at a minimum. I have a 6-stall barn because I don't like feeding and grooming out in the rain. On the upside, we have lots of grass turnout which is awesome, but in the winter it turns to mud so you need to have a dry lot to spare your horses' feet and your fields. Do some reading on good pasture management for your area. Usually you can find info through any local universities that have agriculture programs. The amount of turnout will play a role in how much you feed. With ponies you will need to be very careful about their weight. Most get fat if they so much as look at a pile of hay and this can be extremely harmful to their health. Because of their small size and tendency to be overweight they are more prone to foundering than larger horses. Pasture fencing needs to be horse safe, no barbed wire, no sharp edges to get caught one, etc. There are many options at many price points. I have three rail vinyl fencing at home and at the barn I board my show horse at. My neighbors have high tensile coated wire horse fencing, you could go with 3 or 4 rail wood fencing. Check out your options and see what will fit in your budget while also being safe.
Cost to keep them... It will depend on your area. Beware, horses (and ponies) are extremely expensive and the bills can add up in a hurry. Ponies in general are easy keepers and won't need a whole lot of hay but it will depend on the individual. Most probably don't need any grain but if you end up with a hard keeper you may need to add that into your feed bill. Here, hay is around $300 a ton. I have a larger retired horse and a mini and we go through about 3 tons per year. I would count on needing 2 to 3 tons per year for two average sized ponies to be safe when considering whether or not it is affordable for you. The farrier will need to do their feet regularly - plan for a farrier visit every 6 - 8 weeks. The two I have at home get barefoot trims which are $40 per horse. My show horse has a full set of shoes from a farrier who has more experience working on sport horses, his are $120 every 6 weeks. You'll have to check prices in your area as they vary greatly depending on where you live. Ask the barn owner/trainer where your daughter takes lessons or at a local tack shop and they can refer you to a qualified farrier. Proper hoof care is vital to the soundness and health of your horse or pony so this is not an area to skimp. Routine vet care for my horses consists of routine vet visits twice per year to do shots and check teeth. Shots and vet visit usually run me around $200, if we need to do teeth as well it is around $300 - $350 depending on whether I have to do one or both. Be aware that you will inevitably have additional vet visits. Horses have a knack for finding a way to hurt themselves even in a padded stall and vet bills from injuries, colic, or other illnesses can run you thousands upon thousands of dollars. Also, If your kids end up wanting to show you will be investing a small fortune in clothing, lessons, trailering, etc. Prepare yourself
Horses/ponies are a lot of work. They need feeding at least twice a day, every day. They need stalls cleaned every day, pastures at least weekly. They need adequate exercise, farrier, and vet care and you will have a much harder time going on vacations. That said, if those are sacrifices you and your family are willing to make you will find them
incredibly rewarding. I got my first pony at 6 years old and I wouldn't trade them for the world! They have brought me so much joy and really taught me a lot about responsibility as a kid. Good luck on this big decision, keep us updated on what you decide to do!
