My girlfriend actually used to make a living training horses (and training people to ride them). Here are her comments on this thread:
Horses + Terrain:
(She adds: Nowdays I'm an ancient historian who studies the use of cavalry in ancient campaign warfare.)
This depends on 4 factors -
1. The size/breed of the horse.
2. The weight of the rider.
3. The type of terrain.
4. Whether or not the horse has shoes.
Regarding 1: Large heavy horses will do worse all around. This not only includes both densly wooded areas and marshy footing, but on smooth not so rocky terrain. Lighter/non-draft horses will always be lighter footed and faster in any sort of terrain, simply because their natural coordination will be more reactionary. Ponies, mules, donkies, onagers -- they're all going to be strong, but not quick.
Regarding 2: If you figure armor even in it's lightest state is on average going to add 80 pounds to a rider, this is a problem no matter what sort of terrain you're on. Horses are already hindered with people on their back - they cant make natural choices about direction and timing. If any sort of equipment, saddle bags, or weapons are added to this mess, it's really going to slow the horse down. Rocky terrain is a no-no here. The horses hooves and legs will take a royal beating after a very short time. Endurance riders typically carry very little extra weight and are light people for this reason. Horses cannot naturally keep up high speeds for long periods of time (a Quarter horse is called a quarter horse for it's speed at a 1/4 mile).
Regarding 3: Woods or no woods is not the issue here, let me break it down for you: Swamps are pretty much out for horses. Ever see The Never Ending Story? Yeah. That. Horses in general are NOT big fans of water and have to be trained to deal with it. Even in the wild, they only swim when they have to. So if you've created mud -- you're REALLY creating problems for the horse. Forests are really no different for them as long as it's not dense forest. Bigger horses, in a way similar to elephants, can just kindof ram through the brush, but this takes a lot of energy and really causes their legs and lower bodies to take a beating. If the forest has solid ground, it hasnt rained recently, and there's not a lot of viny underbrush the horse will do just as well here as it would on a plain. (That is, without any sort of heavy weight bearing armor/equipment). Figure the horse will take some light damage in need of tending to from crashing through brush. Really - it's rare a horse makes it through the woods without a scratch on it, but it can make it through them, and can do it quickly with only light damage. As far as plains or rocky plains go, lighter horses will always have an advantage in speed, though not nessecarily in footing. (Smaller hooves = less balance = more likely to fall if they take a wrong step.) Rocky plains will always do a number to horses feet - particularly if they are not shod - that is, they have no shoes. Mountains are hard on horses no matter what. They will move faster than a human would unless the incline is above a 30-40 degree angle. Once it get's beyond that though, they have a SERIOUS problem. Their center of gravity doesn't exactly help in this situation and often times horses will refuse to go up an incline higher than that all together. Futhermore, keep in mind that horses are reactionary in a herd mentality -- if someone else's horse refuses or spooks, there's a good chance that your horse will as well. Finally, as far as indoor riding goes: banish all thoughts of Gandalf and Shadowfax charging up the stairs here. Horses do not do well on marble, or on asphalt. Anything that is rock solid a horse get's no traction on, and must be specifically trained to deal with. Even then it's really dangerous for the horse, and if the horse falls, it will undoubtedly be injured in such a way that will have it laid up for months if not need to be put down. (Think high level spell here.)
Regarding 4:
Shoes are a big point of contention. A lot of horse people say horses dont need shoes...but there's a reason the Roman's invented them. The more armor/weight you put on a horse, the more support it's feet need. The more injuries it can take going over different terrains. Horses dont carry people on their back in the wild....this is ALWAYS an unnatural state. If the horse doesnt have shoes, it's going to take a beating on it's feet. If a horse has to travel more than 100 miles, at decent speeds and doesnt get proper hoof care and rest, the horse will go seriously lame. If it's not rested, it will go lame. Also keep in mind - horses are not cheetahs with a mane and tail, a neat snuffy nose and hooves. Big horses are slow (think drafts), little horses are quick (think Arabians, Thoroughbreds). They are also hot, cold and warm blooded (no not like lizards). Hot blooded horses have drier hooves, do better in desert regions or dry open plains and are the fastest, but will be sluggesh in winter months. Cold blooded horses are your quintessential midieval horses, they do well in colder regions, but were built to pull and charge, not be quick. Their hooves will dry out very quickly in hotter regions and they will take more damage that way. Warm bloods are a combination of the two, they do best in warmish climates, but their hooves will take a beating at either very hot or very cold climates. To really understand how the horse works with it's feet you have to have an understanding of how it moves.
The point being here...that horses are a big can of worms to open up if you want to do it right. Otherwise, it's probably not worth your time. Horses are grazing animals in the wild, they run and deal with what they have to when they have to, but the rest of the time remain at rest. The minute you put them in a situation with people and armor and weight, everything changes - food intake, water intake, hoof wear, teeth care...et al. Less you're thinking about introducing equine colic to your next campaign...I recommend steering clear of all this.