Most of the time, yes, grounds should be connected together but without seeing your schematic it is not possible to say if there are any special considerations that may affect exactly how your ground connections should be made on the PCB.

Same for your supply rails.

Good rule of thumb is: 1) Route all other signals except GND. 2) Make a copper plane that's signal net name is GND and deposit that copper plane to top or bottom. 3) If there are GND pins/pads left that copper plane was not able to reach then connect GND trace to those leftover pins/pads manually. Regards, Markus_ee