Fourier transform vs Fourier series

The set of periodic functions is surely a subset of the set of all functions (most of which are aperiodic). So the Fourier series – which allow us to describe periodic functions differently – are a "subset" (in this sentence, the word "subset" is stranger than before) of the Fourier transform – which is a tool to encode an arbitrary function.

Alternatively, we may see the "subset" relationship in the frequency representation, too. Fourier series may be written as a special case of the Fourier transform in which only the frequencies that are multiply of the basic angular frequency $2\pi/T$ where $T$ is the periodicity are allowed. The Fourier transform of a periodic function is a very special kind of a function, a combination of delta-functions $$\tilde f(\omega) = \sum_{n\in{\mathbb Z}} c_n \delta(\omega-n\omega_0) $$ and functions that are a combination of delta-functions like that (determining Fourier series) are a subset of all functions, including distributions (which may be identified with the Fourier transform of the original function).

I would say that neither is more general, they're different things.

Fourier transform can be defined in a very abstract way, for arbitrary locally compact abelian groups, and sometimes even more general objects. In case of locally compact abelian groups, it turns a function defined on a group into a function defined on its dual, and the formula is quite simple: $\hat f(\xi)=\langle f\mid \xi\rangle=\int_G f \cdot \overline{\xi} \, d\mu$ with $\mu$ the Haar measure.

Fourier series are the instance of Fourier transform where the group considered is the circle group $S^1={\mathbb T}$ (or its finite power), and it so happens that its dual is the group of functions of the form $e^{ikx}$, isomorphic to the group $\bf Z$ of integers (and the Haar measure is just the counting measure); in that way, Fourier transforms are far more general, but that's probably not what you meant to ask about.

What you know as Fourier transform (and what is most commonly referred to when Fourier transforms are invoked) is probably the Fourier transform of a function on the additive group of real numbers $\bf R$ or ${\bf R}^n$, which happen to be isomorphic to their duals, consisting of the functions of the form $e^{i\xi x}$.

I'm not quite sure I understand what you're saying about periodic versus non-periodic pulses, but you can think of a Fourier transform as a projection of a function onto a basis of functions. The same way you can project a vector onto the x, y, or z axes, you can project a function $f(x)$ onto the sine basis or the cosine basis. The Fourier transform projects functions onto the plane wave basis - basically a collection of sines and cosines. A Fourier series is also a projection, but it's not continuous - you sum over $\sin(nx) + \cos(nx)$, whereas in a Fourier transform you preform an integral - a continuous sum.

$$ \int_{-\infty}^{\infty} \;dx\; e^{ikx} f(x) = \int_{-\infty}^{\infty} \;dx\; (\cos(kx) + i\sin(kx)) f(x) \quad\quad x\in\mathbb{R}$$

I think it makes sense that a Fourier series is a subset of a Fourier transform, as the transform takes into account all the same sines and cosines that the series does, but adds in all the noninteger sines and cosines as well.

Hope that helps.