How are subshells and shells arranged in an atom?
I will start off by offering an apology for anything I might have misunderstood - if this is the case, just tell me in the comments below.


  If 2s and 2p are in the second energy level, and 3s, 3p, and 3d are in the next (3rd) energy level


This is, in general, not the best choice of words. The 2s and 2p orbitals are only degenerate, meaning that they have the same energy, in a hydrogenic atom: a species which contains one nucleus and one electron. The hydrogen atom itself satisfies this requirement. Other hydrogenic species include $\ce{He+}$, $\ce{Li^2+}$, $\ce{Be^3+}$, ...

For any species that has more than one electron, the 2s and 2p orbitals have different energies (see this question, although it's a bit of a read) and therefore it would not be correct to refer to them belonging to the same "energy level".

What you can say is that 2s and 2p orbitals belong to the same principal quantum shell. All this means is that the number in front of them - called the principal quantum number - is the same, in this case, 2.


  How are these subshells arranged in space in relation to one another?


Orbitals are not actually containers, or tangible objects that occupy space. They are mathematical functions obtained by the solution of the Schrodinger equation. The pictures of orbitals that you are probably familiar with are just pictorial representations of these functions. Without going into too much detail, one quantity that is used in the functions is the radial distance between the electron and the nucleus, denoted $r$. $r = 0$ corresponds to the nucleus's point in space. What this means is that all orbitals - regardless of their shape or energy - are centred at the nucleus. The $\text{s}$ orbitals are spherical in nature, and the centre of the sphere corresponds to the nucleus. And the same goes for $\text{p}$ orbitals, $\text{d}$ orbitals, and so on.

Therefore, to put it very succinctly, everything is superimposed on everything. You could think of it as shining different colours and shapes of light onto a screen - the projected images can all simultaneously exist.


  3s, 3p, and 3d orbitals all of a relatively larger size


You're right about this - in general, orbitals with a larger principal quantum number tend to be larger, as long as we restrict the discussion to the same atom. For example, the $\text{3s}$ orbital is larger than the $\text{2s}$ orbital, which is in turn larger than the $\text{1s}$ orbital.

Now since I did say that orbitals are mathematical functions, this does raise the issue of how you define an orbital to be larger than another. If you are interested, you can read up about the radial distribution function, although I shall not expand on it as that would really be beyond the scope of this question.