I agree that this may be a better fit for Worldbuilding than here as it is rather speculative.
Answering your question directly, things traveling about in our solar system do so very quickly. Earth travels through space at 30km/s. Near-Earth objects travel in excess of 25km/s. The speed of debris from chemical explosives, however, is usually less than 8000m/s -- and even less for a nuclear explosion, apparently.
For all intents and purposes, the debris from a space battle will expand spherically. Double the radius of a sphere and its volume increases eight times. I haven't worked it out, but I don't think it's going to take long for the debris from this battle to approach that of the nominal density of the interplanetary medium, usually given as 5 "particles" per cubic centimeter, but possibly as high as 100 particles per cubic centimeters in some places -- and they would still be traveling slower than objects in some of the most hazardous areas of the solar system.
And that's idealized; not all debris is going to be racing out from the centre of battle at 8000m/s. Some of the debris from battle will be moving quite slowly. It seems obvious that the closer you get to the place where the battle took place, the more dangerous it is going to be and probably the more armoured you're going to need to be to avoid damage. These areas of battle would probably be designated as some kind of navigational hazard or "no-man's land".
Depending where the battle took place, I suppose, the gas giants would tend to sweep up a lot of that debris, as well.
Having said that, space battles don't seem to be a very realistic option for a number of reasons.
Just as on Earth, I don't think it's likely that great fleets are going to meet each other in open space any more than they would on the open ocean here on Earth. It has happened, but is quite unusual. To engage a fleet you need to know where it is. It's easy to know where a fleet is harboured, and its easy to determine where it may attack, but unless there is some kind of navigational "pinch point" (like a cape, a horn, a Northwest Passage, a Rock of Gibralter, a Red Sea, etc.) it may not be so easy to know where it is while in transit -- especially when it is actively attempting to conceal both its position and its strategic timing.
For that reason alone I would think that most space battles are going to occur close to known bodies, but there is another interesting consideration. The things which make fleet battles possible here on Earth do not exist in space. Moving about in space is not really very similar to movement on a body of water. Space is much larger, for one, and is in three dimensions as opposed to basically two-dimensional waterbourn travel. In space you don't actually travel from point A to point B as you would on Earth because in space points A and B are both moving! You leave point A and then you have to actually try to meet point B as it goes past a known point in space at a certain time, and if you miss it then you're basically pooched!
And figuring all that out is not a reversible calculation! Knowing when to leave point A in order to meet point B does not tell you when you would need to leave point B in order to meet point A!
But let's say that you have some kind of torchship technology so that you can travel to any arbitrary point in interplanetary space and don't have to worry about fuel and spatial rendezvous and such. The problem here is that even interplanetary space is very big! To get anywhere within a half-way reasonable period of time you need to travel very fast all the time. Unless both your fleets just happen to be traveling in the same direction at similar speeds, in order to meet your enemy your fleet would need to accelerate in his direction, then decelerate to engage him in battle. If your enemy chooses not to decelerate to meet you then he would simply just blow straight past you and attack his strategic target of interest.
In short, whether you have torchship technology or not, the navigational problems are quite a bit more complex than just taking your ships out to oppose an attacking fleet.
For all these reasons I would conclude that for any space battle to be fought at all it would probably need to be fought in the vicinity of the target of strategic importance; to wit, close to or within the influence of a gravity well. Most space debris from these battles would, therefore, be confined to the areas around those points of battle.
But a space battle in the vicinity of a moon or planet basically makes using it very hazardous. NASA says they are currently tracking about 23,000 pieces of man-made debris larger than a softball in orbit around the Earth. They say that there are half a million pieces of debris which are the size of a marble and larger, and about 100 million pieces of debris smaller than a marble. I think that one or more space battles would put so much debris in orbit in the vicinity of a planet or moon that you would need -- as you've already said -- a lot of armour; and a lot energy to land such a heavy vehicle, not to mention getting it back off the ground again.
Space battles do seem to be a little problematic.