Which force field can increase a moving electron's speed?
An electric field could.
In theory the gravitational and electric fields could accelerate a moving electron but practically speaking for most cases only an electric field will accelerate the electron noticeably. The magnitudes of the electric and gravitational forces are a product of the charge and mass respectively with some other values.
The charge on an electron is
Electric force - Coulomb's Law:
Gravitational force - Newton's Law of Gravitation:
The magnetic force will not change the speed of a moving electron because the magnetic force is always perpendicular to the velocity. A moving electron in a uniform magnetic field will undergo uniform circular motion.