From discrete to continuous - why quantum *fields*? From what I understand, combining special relativity with quantum mechanics requires quantum field theory (QFT). Why fields?
I can see that for quantities already described by classical fields, such as the electromagnetic field, the quantum analogue will be some sort of field, but what is the motivation for describing particles in terms of excitations of underlying quantum fields?
A possible line of reasoning
Perhaps we wish to describe the creation and annihilation of particles, and perhaps this can be done through introducing creation and annihilation operators that act on the vacuum state. Such operators should act at each point in spacetime since particles can be created at each and every point in spacetime. Furthermore, such particles should be able to propagate continuously through spacetime and so the object that describes them should "operate" at each point in spacetime. Hence we should consider a field of operators, a so called quantum field?!
 A: I think to get a real satisfying answer to this question, you should read parts of Weinberg, vol. I, especially chapter 4, where at the beginning he writes 

[I]f we express the Hamiltonian as a sum of products of creation and annihilation operators, with suitable non-singular coefficients, then the $S$-matrix will automatically satisfy a crucial physical requirement, the cluster decomposition principle, which says in effect that distant experiments yield uncorrelated results. [...] In relativistic quantum theories, the cluster decomposition principle plays a crucial part in making field theory inevitable.
  There have been many attempts to formulate a relativistically
  invariant theory that would not be a local field theory, and it is
  indeed possible to construct theories that are not field theories and
  yet yield a Lorentz-invariant $S$-matrix for two-particle scattering,
  but such efforts have always run into trouble in sectors with more
  than two particles: either the three-particle $S$-matrix is not
  Lorentz-invariant, or else it violates the cluster decomposition
  principle.

The rest of chapter 4 is about supporting these statements and is of course far too long to repeat here.
Weinberg also talks about this in "What is quantum field theory, and what did we think it is?"
A: There is a reason other than the one you cited for picking fields. In QFT a particle is just a state in which the field has a particle number of 1 and n particles are just a field state in which the field has particle number n. In general, there is no such thing as the same particle over time, just changes in the patterns of particle numbers (and other field observables) in fields that propagate locally.
Suppose you have a relativistic quantum theory in which each particle was distinguishable from the others. The state of any such particle would in general spread outside the light cone. So if you measured the particle and found it in region x, it would not be found in region y that is spacelike separated from x. The physicists who found this out found it unacceptable and thought it might allow FTL communication. There are other problems solved by QFT: see Part III of 'Quantum field theory for the gifted amateur' by Lancaster and Blundell.
