Would cavities of near vacuums in the head (or in the 6 x 6 blocks surrounding the rocket [for reentry temperatures]) of a rocket make it lighter so it can be sent out easier?
No because with practical materials, the structure needed to support a vacuum is heavier than the air it displaces. Here's a description of the principle https://en.wikipedia.org/wiki/Vacuum_airship
Even if you increase the buoyancy, say with a helium balloon, you're still not decreasing the mass, and mass is what's expensive to accelerate with a rocket.
In a word, no. Getting into space is a matter of achieving enough velocity to overcome the Earth's gravitational influence, not about buoyancy. You can use buoyancy to get to the edge of the Earth's atmosphere, not beyond. We don't fill a rocket with helium so it'll float into space, nor do we use balloons to get beyond the edge of the Earth's atmosphere.