In this series, the characteristics of intramolecular vibrational-energy redistribution (IVR) present in the CH overtones of CHF3 are investigated. Particular attention is given to the multiple time scales and thus mechanisms present in the IVR dynamics. In Part I, a 9-dimensional ab initio potential energy surface is developed to adequately account for the vibrational couplings of all modes. Furthermore, all-mode vibrational state calculations, of large-primitive space dimension, are performed using a recently developed wave operator sorting algorithm in tandem with the recursive residue generation method. All fundamentals, first overtones, and bimodal combination states with up to 3 quanta are presented. Also, the A(1) and E-symmetry CH polyads are determined through the second overtone. Equilibrium geometry, rotational constants, and vibrational properties agree quantitatively with experiment in most cases. The error is systematic in origin and largely due to the error in the ab initio harmonic frequencies. New vibrational constants and resonance interactions are reported for the background modes. In contrast to the prominent CH stretch-bend Fermi resonance structure, responsible for ultrafast (t<50 fs) energy transfer, the CH polyads also exhibit vibrational fine structure of order I to 10 cm(-1) due to background-mode coupling. This secondary coupling results in IVR on the picosecond time scale.