Do I need a ground plane?
Whether or not you need a ground plane depends on the frequency of operation of your circuit (which you have not told us). If it's a low-frequency circuit (say, audio frequencies), then you probably don't need a ground plane. But if it's a high-frequency circuit (say, 1MHz or above), then you do need a ground plane.

As the commenters have said, you should make your power and ground traces wider than your signal traces. And since it looks like you have room on the PCB, you should make a ground-fill area, even for a low-frequency board. This means that all the bare PCB area will not be bare insulator, but covered with copper and connected to ground.