  In this paper and its sequel, we study non-equilibrium dynamics in driven
1+1D conformal field theories (CFTs) with periodic, quasi-periodic, and random
driving. We study a soluble family of drives in which the Hamiltonian only
involves the energy-momentum density spatially modulated at a single
wavelength. The resulting time evolution is then captured by a M\"obius
coordinate transformation. In this Part I, we establish the general framework
and focus on the first two classes. In periodically driven CFTs, we generalize
earlier work and study the generic features of entanglement/energy evolution in
different phases, i.e. the heating, non-heating phases and the phase transition
between them. In quasi-periodically driven CFTs, we mainly focus on the case of
driving with a Fibonacci sequence. We find that (i) the non-heating phases form
a Cantor set of measure zero; (ii) in the heating phase, the Lyapunov exponents
(which characterize the growth rate of the entanglement entropy and energy)
exhibit self-similarity, and can be arbitrarily small; (iii) the heating phase
exhibits periodicity in the location of spatial structures at the Fibonacci
times; (iv) one can find exactly the non-heating fixed point, where the
entanglement entropy/energy oscillate at the Fibonacci numbers, but grow
logarithmically/polynomially at the non-Fibonacci numbers; (v) for certain
choices of driving Hamiltonians, the non-heating phases of the Fibonacci
driving CFT can be mapped to the energy spectrum of electrons propagating in a
Fibonacci quasi-crystal. In addition, another quasi-periodically driven CFT
with an Aubry-Andr\'e like sequence is also studied. We compare the CFT results
to lattice calculations and find remarkable agreement.
