I've joined several forums for historical novelists recently while I've been completing COACHMAN. Reading some of their book previews on Amazon it has dawned on me that while historical novelists seem interested in being authentic about costume and weaponry, I've only found two who have any direct knowledge about horses, and one who understands carriages.
Given that until the start of Victoria's reign (or William IV if you are a Liverpudlian) the fastest means of transport anywhere was either a well bred horse or a well-horsed carriage, this seems rather strange. It's a bit like writing about modern London and not knowing that you can get your character across the city far more easily by Tube than by Ferrari.
Allow me to make some helpful information available!
Even for late medieval knights, war-horses were not huge beasts. (The modern Shire horse of 17 hands or more was developed for agricultural use, well after gunpowder put an end to the mounted armoured knight.) Ann Hyland proposes that the 15th C knight's "destrier" was likely to have been a sturdy cob type of about 15-2 hands. This type of horse was a "square gaited" animal who walked, trotted, cantered or galloped.
Horses for general riding were probably not taller than 15 hands, often smaller, and frequently they were "lateral gaited" or "soft gaited", which means they walked and cantered but their in-between gait was a pace, rack or amble, rather than a trot. All these were more comfortable than a trot; the pace is similar to that of a fast moving camel or giraffe, with the legs on each side moving at the same time (watch videos of harness-racing pacers). The amble was a rapid, level, shuffling movement. Over good going the rack, like the pace, could be quite fast. Horses that moved in this comfortable way were amblers or palfreys, popular mounts for both ladies and gentlemen.
Horses for racing or hunting were "coursers", ie running horses, who moved "square" like the knight's destrier; they trotted when not galloping, and even the racehorses were not large. Many early thoroughbreds were under 14 hands high.
A useful type from Shakespeare's time onward was the Galloway, originally a type from south-west Scotland. Later the term became a generic one (like Hoover for vacuum cleaner) for any sturdy, sensible small cob of 13 to 14-2 hands. They probably looked much like a Highland, Dales or Fell pony, though for practical reasons they probably didn't carry anywhere near as much hair on neck, tail or fetlocks ("foot locks") as their modern counterparts do.
Women travelling would have a choice of riding pillion behind husband or brother; rich ladies could ride side-saddle; or ride in a litter, which was a covered seat carried on 2 long poles, suspended either side of a horse before and behind.
Modern endurance horses ridden intelligently at suitable paces for the terrain can cover anything from 25 to 50 or even 100 miles in a day. This isn't just the racy Arabian types; I know a Fell pony that has completed a 100 miler in less than 18 hours. Bear in mind that the riders don't flog on at full gallop, which being anaerobic activity would rapidly exhaust the horses' energy; they go at the pace that suits the country they have to cover, and much of that is aerobic, steady trotting or cantering, with intervals of walk to allow the horse to have a breather.
Military dispatch riders galloped, of course, and covered short distances of 3 or 4 miles in a quarter of an hour (12 to 16 MPH); but ridden horses for civilian travellers over longer distances would average no more than 6 MPH, so a day's journey was probably no more than sixty miles and often much less if roads were bad. Working horses in a city were often forced to walk because of other traffic, and there were few highway regulations to ease their passage.
Unlike cars, horses do need recuperation time before they can reasonably be expected to make another journey. This involves them being groomed, fed with grain and hay, and given water to drink and a stall under shelter in which they can stand quietly and perhaps sleep - though horses can sleep standing up and tend only to lie down for short periods.
Part 2 will look at travelling by carriage and coach.