Since we want to have a lot of surface area built quickly, we need to find a way to minimize the amount of time and resources to build spaceships and associated systems. To minimize energy costs most of the mining, processing and construction will take place in space. This can be done on the Moon, or using the resources of Near Earth Objects (NEOs), although for speed it may be necessary to use multiple sources of materials.
For the main bodies of the spacecraft, the formula will be simple. Two spheres of metal, with the inner one being five metres in diameter smaller than the outer one. The interspace is filled with water mined from the Moon or NEOs, which provide radiation protection, thermal buffering and basic reserves for life support. The spheres can be even as thin as heavy foil reinforces with a close knit netting to reinforce it, but this assumes the interior will always be in microgravity. A sphere provides maximum volume with minimum surface area, and thus minimal material budget.
Using mirrors to bring sunlight into the interior, and tapping into the vast reservoirs of water provides the means to grow algae. This provides a means to consume and recycle wastes, provide the basis for engineered foods and to consume the CO2 in the atmosphere and provide fresh oxygen. The algae bioreactors can be as simple as clear plastic tubes exposed to the light, although the entire suystem is obviously much more sophisticated in handling inputs and outputs.
Even propulsion can be relatively simple, using photonic laser thrusters, which essentially uses powerful lasers on the Moon or other bodies to beam at a mirror on the spacecraft, then reflecting the beam back to the base station to re reflect to the mirror - essentially a highly evolved form of solar sail.
Much of this requires building some basic infrastructure on the moon and then leveraging it. Rather than building a single huge spaceship, it would provide more security and redundancy to build multiple smaller craft. This also provides more flexibility in that you can disperse the population around multiple different moons and planets around the Solar System.
Without specifying things like the population, desired population density and so on there is no way to really quantify how much material you will need, but a series of minimalistic spacecraft built along these lines can be built quickly and cheaply, allowing you to move large numbers of people around the Solar System in an economical manner.