Why do we learn only two computations, cross sections and decay rates, in such a fundamental theory as QFT? When I learned Newtonian mechanics I found a vast variety of computations that I could do and that was so interesting. And it was so when I learned Maxwell theory. When I started learning QFT I hoped to find much more variety of computations and it's so disappointing when I see only two! Of course I'm not a fan of computations but every new computation is a new window to understanding the theory and appreciating it's beauty and power.
 A: The small number of "conceptually independent types of processes and calculations" is exactly a symptom of the theory's being fundamental! Even in classical physics, all calculations could have been mathematically reduced to the calculation of the final state that evolves from an initial state (or a state that is stationary etc.).
In quantum mechanics, this must be replaced by the calculations of the probabilities that an initial state evolves to a final state. According to QFT, all objects in the world may be described by a Hilbert space with some particle excitations (creation operators).
All the dynamical transformations are included in the probabilities to transform an initial state of particles to a final state of particles. For the calculation to be nontrivial, the initial state contains at least 1 particle.
If it contains 1 particle, the only nontrivial process that may occur is the decay of the particle. If it contains 2 particles, they may do something and the probability is unavoidably described in terms of a cross section because the probability depends on the flux of the beams etc.
If the initial state contains at least 3 particles, it becomes extremely unlikely - at least in the empty space - that all the particles interact simultaneously. Instead, some interaction of 2 particles occurs first, and that interaction may be reduced to the cross section calculation from the previous paragraph. 
So basically all processes may be reduced to probability amplitudes of the two kinds. This is a big victory of reductionism.
It doesn't mean that QFT doesn't allow one to calculate everything you could have calculated in classical physics – or other theories less complete than QFT. These calculations are just hard and, in some sense, they are not fundamental or elementary.
In classical mechanics, one may compute some behavior of a machine with lots of wheels and gears etc. This is clearly an example of applied, not fundamental, physics, and researchers of QFT generally do not do applied physics.
People usually study relativistic QFT because they want to learn the fundamental laws of physics and that's why they're not deliberately focusing on more complex and composite "exercises". But lots of them are possible. In principle, the behavior of all composite objects may be probabilistically predicted using QFT. And advanced papers using QFT surely do use lots of concepts that differ from decay rates and cross sections, i.e. viscosities (in the AdS/CFT correspondence applied to ion physics etc.) and new "emergent" methods to calculate them.
A: One can calculate a lot more in quantum field theory if one goes beyond asymptotic computations into thermal field theory. I recommend that you look at the book ''Nonequilibrium Quantum Field Theory'' by Calzetta and Hu.
