Usually by truck, unless you are going 3000+ miles or over an ocean. Air shipping is very expensive, and can be very dangerous to the horse (if they don't ship well, they can be euthanized to protect the plane). This is of course a much lower risk with a mini. Truck is pretty normal. 1000 mile trips usually take two days. Some people/companies unload half way and let them rest, others drive straight through. Some go door to door, some work out of hubs. Some allow you to rent the whole trailer to get direct service, most do a circuit loading up, then drive to the destinations. Some companies offer box stalls (my preference) and some do slants/reverse slants. Large rigs (semis) allow box stalls or straight loads, as do some larger goosenecks. Moral of the story is there is a different company for pretty much anything you want. Shipping a single horse can be $0.50-$2/mi, multiple horses varies greatly. I've used All State Express several times. Sometimes I really liked them, others, they weren't as good. But I'd recommend them. I've never worked with any of the companies that specifically ship minis so can't help you there.
Shipping is such a normal part of a horse's life these days that its almost routine. There is almost nothing to worry about with a quality shipper
I've done several 1000 mile drives myself, they are a lot of fun. I put 5000 miles on a truck in a single month last year hauling all over the country.