The standard description of cosmological observables is incomplete, because
it does not take into account the correct angular parametrization of the sky,
i.e. the one determined by the observer frame. This can be accomplished by
introducing a tetrad at the observer point representing that frame. In this
paper we consider the tetrad formalism of General Relativity and develop a new
formalism for describing cosmological observables. It is based on the "observer
space-time" manifold, whose coordinates are the proper time, redshift and
angles an observer uses to parametrize measurements, and on which the rest of
the observables are defined. This manifold does not have to be diffeomorphic to
the true space-time and allows us to resolve caustics in the latter, in
contrast with similar coordinate-based formalisms. We work out the definitions
and equations for the angular diameter distance, weak lensing and number count
observables. As for the ones associated to the CMB, they lie inside the phase
space matrix distribution of the photon fluid evaluated at the observer point
and pulled back on the observer sky. The second part of this paper is devoted
to general-relativistic matrix kinetic theory. Here too the tetrad formalism
appears as the natural approach for relating the macroscopic dynamics to the
microscopic QFT. Part of the presented material is known and is included for
completeness, but we provide more detailed discussions over some subtle issues
and we also consider an alternative construction of the collision term which
deviates from the standard one at higher order in the loop corrections. In
summary, the present paper contains all the required structures for
computations in cosmology with exact and model-independent cosmological
observables. The associated linear perturbation theory will be given in a
companion paper.