A DC motors speed is directly proportional to the voltage applied. Generally speaking if you increase the voltage you increase the speed. The Torque in a DC motor is directly proportional to the armature current, assuming the field is fixed (permanent magnet). I use the following simple equations to help me remember what a DC motor will do in various applications.
Vt = Vcemf + Ia*Ra +BD (voltage equation)
Vt = Terminal Volts
Vcemf = Volts counter emf
Ia = Armature current
BD = Brush Drop
n = Vcemf/N*If*K (speed equation)
n = speed
N = number of turns of wire in field coil
If = Field current
N*If = Field Flux
K = Constant
T = Ia*(N*If)*K (torque equation)
T = Torque
From the Torque equation you can see that Torque it is not dependent on voltage. If you raise the motor volts it will raise the motor speed but not necessarily lower the armature current. The armature current is dependent on the amount of load on the motor. If the application were a fan or pump then raising the voltage would raise speed which could actually increase the load on the motor (more armature current)
There have been a number of other threads where the DC motor equations were discussed in some length. Maybe one of the other people can respond with the links.