>>13456045There is incomprehensibly much living space in our solar system. The asteroid belt alone, comprised of 0,5% of the mass of the Moon, has enough resources and space for quadrillions of humans.
We will almost certainly use light-sails pushed on by a stellaser in order to reach interstellar cruising speeds, sizable fractions of lightspeed, and either a drone-built stellaser at the destination system or some kind of fusion drive to decelerate. It's doable but the scale is very large, it's a problem for the coming thousands of years not for us.