  We study the mixed-type classical dynamics of the three-particle
Fermi-Pasta-Ulam-Tsingou (FPUT) model in relationship with its quantum
counterpart, and present new results on aspects of quantum chaos in this
system. First we derive for the general N-particle FPUT system the
transformation to the normal mode representation. Then we specialize to the
three-particle FPUT case, and derive analytically the semiclassical energy
density of states, and its derivatives in which different singularies are
determined, using the Thomas-Fermi rule. The result perfectly agrees with the
numerical energy density from the Krylov subspace method, as well as with the
energy density obtained by the method of quantum typicality. Here, in paper I,
we concentrate on the energy level statistics (level spacing and spacing
ratios), in all classical dynamical regimes of interest: the almost entirely
regular, the entirely chaotic, and the mixed-type regimes. We clearly confirm,
correspondingly, the Poissonian statistics, the GOE statistics, and the
Berry-Robnik-Brody (BRB) statistics in the mixed-type regime. It is found that
the BRB level spacing distribution perfectly fits the numerical data. The
extracted quantum Berry-Robnik parameter is found to agree with the classical
value within better than one percent. We discuss the role of localization of
chaotic eigenstates, and its appearances, in relation to the classical phase
space structure (Poincar\'e and SALI plots), whose details will be presented in
paper II, where the structure and the statistical properties of the Husimi
functions in the quantum phase space will be studied.
