I got 3 lights above my kitchen counter controlled by a single switch, would I need 3 bulbs, or would the single act as a switch to control the others

You would need 3 bulbs. You might want to consider a smart switch instead.

I would think yes. Since its controlling the lights themselves and not the switch.

You would need 3 smart bulbs. You can then set them as a group and just say turn off counter lights or whatever you set it too and it will turn off all three at the same time.

You would need 3 bulbs. The device communicates with each one to operate the lights...if you only install one light bulb only that one will be controlled by the device.

If you want "smart" bulbs that you can control independently of one another (i.e. turn one on and leave two off) then you would need three bulbs and each would be connected to the Google Home Mini. If you don't to change the bulbs and are proficient with changing out your switch, you could get a smart switch that is compatible with the Google Home Mini and install that. That way the switch is then connected to the Mini and can control all three lights at once. That would leave you free to put the bulb that comes with the Mini somewhere else.