Dear @jremington thank you for your reply and sorry for my imprecise language.
What I mean is that I expect the motor shield to have some kind of PWM pin to which I can send a command like
analogWrite (pwmPin, 127); //
and expect roughly 12 VDC output if the power source is 24 VDC.
I am not sure on how to use the H-bridge to change polarity, I'm still studying. As far as I understood some motor shields have a "Direction" pin that might be set HIGH or LOW and that should correspond to +12 VDC or -12 VDC in the previous example.
I am also not sure on how to combine these in a PID except with some kind of logic control on the temperature value that tells the Arduino that if the temperature is too cold it needs to set LOW the Direction pin before applying the PID output.
Again, sorry if all of this is veri naive or basic, I am really trying to piece together all the information and there's a -lot- of it.
Anyway, as you suggest, I'll keep checking this forums for good examples