Unfortunately, descriptions that are for more than the simplest conditions are difficult to perceive by someone who has never visited the site. It is almost always true here that a picture is worth 1,000 words. It would be best if you add a few pictures so that people here know how this yard lays out and where you're going with what. For proposed elements, a plan in the form of a reasonably accurate sketch would be helpful, too. But start with the pictures.

You've said, generally, what you plan to do with water in pipes, but you've not made any case for why you think it's necessary. Water in pipes frequently brings trouble (especially if not planned to a "T.") In many cases, water can flow above ground with much less trouble (and much less effort and expense than installing an underground pipe system.) It depends on the flow, but often plants do a better job of protecting against erosion than rocks do. Often, rocks and concrete become "in the way" (and an expensive, troublesome element to rearrange or dispose of) at a future date when a change is deemed necessary. You might want to explore options before committing to the proposed drainage plan.

Thanks for all the info.

I get what you are saying about water flowing above ground, which will work for the downspouts facing the back, but two face the front yard, which, because of the grading done when the house was built, means the water would probably just run into the street instead (there's a very slight slope down towards the road in the front yard)...is there any way to make sure it at least starts its journey "facing" the right direction without using pipes?

If the grading is proper, there should be a downhill pitch away from the house all the way around it for a distance of 10', if available. In many cases 10' will not be available (like where houses are only 10' apart) so one works within the confines of what is. Once water is away from the structure, grading will proceed according to site conditions, with the goal of allowing it to leave the property in the most direct, unobstructed route, taking into consideration the many obstacles it may encounter. Grading routes it around the immovable obstacles.

Whatever is the general lay of the land, the drainage will work WITH, not against it. At the front yard, if from the front of the house the grade slopes downhill toward the fronting street, then water coming from the front downspouts will flow toward the street ... not toward the back yard or back lot line ... even if, from the house, the back yard slopes downhill toward the back lot line. The back yard downspouts will flow toward the back lot line, not toward the front street.

The downspouts end in a tail piece that directs water away from the house and then feeds into a splash block made of concrete (better) or plastic (less good) to be guided another 20" in the correct direction. (Sometimes an extension to the downspout tail piece may be required.) Since water leaves the splash block in the downhill direction, it will keep going in the same direction. If the landscape is good, the ground at the splash block will have some sort of planting (grass or groundcover usually) that will protect the ground from the forces of fast moving water and erosion (another reason why mulch should be a temporary condition, not the permanent fix.) If there is not good planting around the splash block, the fast moving water will usually dig a hole at the toe of the splash block, undermine it, and disrupt earth or mulch in the path of the exiting stream. Good planting solves all of these problems.

use rigid PVC and cleanouts where you have bends or Ys and you'll be fine. All my front downspouts are captured with 4" PVC S&D pipe and directed into the backyard, seven years with zero issues,