On Foot
Although a runner might travel much farther and faster, the run-of-the-mill person can walk just over three leagues (10 mi) in a day.
In hilly terrain, the cost in travel time increases by fifty percent -- so that a person moves about two and a half leagues (7.5 mi) in a day. In mountainous terrain, travel time doubles, reducing speed to one and two-thirds of a league (5 mi) in a day.
On Horseback
The primary benefit of owning a horse is that you reduce the time it takes you to travel. Just how much of a reduction? Several factors go into this consideration: the horse's breed, the horse's age, the terrain, how it's trained, how it's fed, and whether or not it's encumbered.
The very least one would get out of a horse is just under 7 leagues (20 mi) in a day, unless it was pulling a cart. A well-conditioned horse in its prime years, on good feed and without being heavily encumbered, would be able to comfortably travel 16 and a half leagues (50 mi) in a day.
Terrain, of course, affects horses as well as humans. A horse unused to hilly terrain will be 30% slower. A horse used to hills will be only 20% slower. On a steep slope, it is recommended that the rider dismount and lead the horse. Traveling quickly through a dense forest is feasible, but sometimes risky. And in case it needs to be said, swamps are not suitable places for horses.
Take heed when purchasing a mount that there are many different breeds of horses, and always remember to take care of your mount by regularly visiting an ostler or farrier and supplementing its grazing with oats as needed.
On Cart or Carriage
Unless it's an extremely light carriage -- such as a curricle -- do not count on a speed advantage just because the vehicle's moving under the power of a horse or ox. A cart or carriage's advantage is not in its speed, but in its capacity. However, a team or at least a pair would make better time with a carriage or cart, if speed is a concern.
On River Raft
As may be surmised, river barges make better time downstream than they do upstream. They can get downstream at a rate of 10 leagues (30 mi) a day, whereas they move only about 5 leagues (15 mi) a day laboring against the current.
On Ship
One doesn't have to be a sailor to know that there are ships built for speed and ships built for hauling cargo. A Korinthosi or Yulese galley is faster than a Ferraran caravel, a Ferraran caravel is a faster (and more maneuverable) than a Ceredigic carrack, and a carrack is faster (and bigger) than an Abydosian cog.
One thing a sailor does know that might surprise a landlubber is that smaller ships sail more slowly. So, a 50' cog's average speed is around 5 knots. A 75' cog's average speed is around 6 knots. A 100' cog's average speed is around 7 knots.
Size and shape are not the only determiners of speed at sea. Under good but not ideal conditions -- that is, when there are days when virtually no progress is made (or even when the ship goes backwards) but also days when wind, tide, and course all come together and the crew logs 150+ miles between sun-shots -- a carrack might do sixteen to twenty-three leagues (50-70 mi) a day, a caravel might do twenty-one to twenty-eight leagues (65-85 mi), and a cog might do thirteen to twenty leagues (40-60 mi). (A galley, having both oars and sails, would be less dependent on wind conditions.)
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