How do I make multiple connections to a microcontroller's GND/VCC etc?
Once you have verified your circuit on a solderless breadboard, a veroborad daugtherboard might indeed be a good next step. I prefer the veroboard that has sets of three holes connected. (others have entire rows connected, or copper rings about individual holes)

A dedicated PCB would of course be perfect, but it takes a lot of effort and/or one-time costs.