Hello Phaeous. Welcome to the forum.
I assume the title is supposed to be included in your text.
The arrangement of fissile materials, moderator, and structural materials in a reactor is chosen most carefully. You need the reactor to be mechanically strong enough to safely support the various forces such as the weight of the various reactor components, the pressure and motion of the coolant, motion of reactivity devices, etc.
But primarily, the material is arranged such that neutrons are slowed to thermal speeds and then arrive at fissile material with the fewest lost that can be achieved.
If a neutron starts at typical energy from a fission, and is traveling through water, it will scatter some few times, probably about 12 times, before reaching thermal energy. During this travel it will move some distance, maybe as much as a meter or two. Probably quite a bit less. I would have to look up the average. And it will depend on the details of the reactor.
But that gives you some idea of what is most efficient. The usual reactor design is fissile fuel arranged in rods in a background of moderator. Other designs are pellets of fuel, plates of fuel, etc. But the usual pattern is an array of small-ish chunks of fuel in a background of moderator.
https://en.wikipedia.org/wiki/Nuclear_reactor
So the neutrons bounce around in the moderator. They slow down. And enough of them find the fissile material in the fuel rods to keep the reaction going.
Neutrons can find other things besides fuel and get absorbed. For example, iron is a neutron sucker. Thermal neutrons that find iron parts of a reactor tend very strongly to get absorbed. This is a problem because the absorption releases a gamma (a photon) that heats the surrounding material. So reactor design tries to keep the iron parts to as little as possible. Some other things that happen when a neutron hits structure components include such things as making it brittle. Or causing the molecular bonds to become dislocated so the structure stretches under force. Pressurized components tend to balloon under neutron radiation.
As well, as you mentioned, neutrons are radiation. And they induce radioactivity in materials that absorb them. And reactors produce other kinds of radiation besides neutrons. All of these are a challenge. They must be accounted for and designed for. This is a challenge, but it is well understood how to do it.
http://thesheaf.com/2010/11/04/u-of-s-wants-to-build-nuclear-reactor/
The picture shows the NRU reactor in Chalk River. It typically operates with 130MW of power. I have stood on the upper deck plate in the picture while it is operating. Quite an experience. I was there to help with data analysis of an experiment, and the technician invited me to observe while he made a measurement. And I was standing there watching. And suddenly I realized, my feet were unpleasantly warm. And the tech turned to me and said "You're feet are hot, aren't they?" And indeed they were since the upper deck plate is 70C during operation. But the radiation at the deck is very little, such that they used to take school children on tours. And they stopped because of 911 paranoia, not the very small radiation.