The grounds need to be bonded to the house ground, per NEC code.
If you are in a lightning prone area, Ground each tower leg to it's own ground rod, then ring all the ground rods together using a heavy cable. Ground the outer shield of the coax near the top of the tower and near the bottom of the tower.
Install a lightning arrestor where the coax enters the house.
Ground the radio and all your shack equipment.
Again, all grounds need to be bonded together.
Might sound like overkill, but that is what you'd see at a commercial site.
At minimum (this won't meet any codes) you need at least one ground rod at the base of the tower and bond to the house ground. Lightning arrestor before coax enters the house.
Ideally you want to give any energy a safe path to ground that doesn't involve passing through your radio. A direct lightning strike will likely do damage even with all of this, although a good installation can survive a strike.
Direct strikes are not the only thing you need to be concerned about. Nearby strikes can induce enough energy into nearby conductors (cable, tower, etc) to cause problems. Also, strong winds and rain/snow can induce static electricity into your system. Give it a safe place to go.