Agfdhyk

A brute force concept using 1970s technology and lunar materials:

Use lunar ores to make reflective materials and Stirling engines. Use these materials to build solar-thermal power plants on the moon. Store the energy using molten-metal batteries or batteries that melt-and-refreeze metals or salts, also made using lunar material.

Use high-power ablation technology for the rockets. The bottom of the ship is a large shaped piece of metal. Send an intense energy beam from the launch site to the bottom of the ship. Boil off the metal, to provide thrust. Earth-based launchers using this concept would need about 3 GW of power. (Per "Halfway to Anywhere", in A Step Farther Out.) Since the moon has about 1/6 of Earth's gravity, 500 MW would suffice.

Bonus points if the "intense energy beam" is a laser beam, mounted on a stuffed shark. (The sharkskin would probably need to be imported from Earth.)

Include a small-scale solar-thermal system on the spaceship, along with a small-scale version of the battery system. Send a modest energy beam from the moon to the ship's solar-thermal system to power the spaceship.

"The Japanese Kaguya spacecraft, which was launched in 2007, detected uranium with a gamma-ray spectrometer. Scientists are using the instrument to create maps of the moon's surface composition, showing the presence of thorium, potassium, oxygen, magnesium, silicon, calcium, titanium and iron."

https://www.space.com/6904-uranium-moon.html

Thus the obvious solution is to build a nuclear fission reactor on the moon and mine for uranium. Nuclear fission (and fusion) do not require oxygen, so no atmosphere is needed on the moon.

Nuclear thermal rockets were prototyped and had (non-flight) tests from the 1950s to 1970s.

To date, no nuclear thermal rocket has flown, although the NERVA
NRX/EST and NRX/XE were built and tested with flight design
components. The highly successful U.S. Project Rover which ran from
1955 through 1972 accumulated over 17 hours of run time. The NERVA
NRX/XE, judged by SNPO to be the last "technology development" reactor
necessary before proceeding to flight prototypes, accumulated over 2
hours of run time, including 28 minutes at full power. The Russian
nuclear thermal rocket RD-0410 was also claimed by the Soviets to have
gone through a series of tests at the nuclear test site near
Semipalatinsk.

The simplest method of obtaining a rocket fuel from the Moon is to mine the ice on the poles. This gives rocket fuel from a single mining source.

According to Wikipedia:

In March 2010, it was reported that the Mini-SAR on board
Chandrayaan-1 had discovered more than 40 permanently darkened craters
near the Moon's north pole that are hypothesized to contain an
estimated 600 million metric tonnes (1.3 trillion pounds) of
water-ice.

Then you just need a bit of heat, a lot of electricity and the ability to compress and to separately store H2 and O2.

You melt the water and use Electrolysis to split the water.*

Wikipedia link for if you don't know what that is:

This technique can be used to make hydrogen gas and breathable oxygen.
As hydrogen is an important industrial commodity, by far most
industrial methods produce hydrogen from natural gas instead, in the
steam reforming process.

Then you just use cryogenic compression to convert the gases to liquid form for storage.

If the ice is not where you want to be launching rockets from, it is easy to transport the ice to the launch area. I would recommend that over transporting the O2 and H2 gases. For one thing, if the transport breaks down, ice will evaporate much more slowly than cryogenic liquid gasses.

*When did they change the term from cracking water to splitting water? A search on "crack water" yielded a whole bunch of links that I wasn't looking for.

Solar power combined with ion engines and a mass driver:

Power Source

Solar power could conceivably be used to power the craft, providing both the onboard power and propulsion.

Solar power is already widely implemented, with solar panels sufficient to supply 227 Gigawatts of electricity having been installed globally by 2015.

The main component of most photovoltaic cells is silicon. This is the second most abundant element on the lunar surface, however, it exists in various ores rather than the relatively pure form used for solar panels on earth. A process to extract silicon from these ores would be required. Such a process has been suggested.

Propulsion in space

For propulsion while in space solar panels could be combined with ion drives - this is a technology which has already been implemented. Ion drives require a propellant, for this a range of elements have been used or proposed, including xenon, argon, iodine, mercury, and bismuth. Designs such as VASMIR could theoretically use practically any material for propellant. Thus it should be possible to find a suitable propellant on the moon.

Propulsion to launch

Ion drives do not, however, provide sufficient thrust to escape lunar gravity. This could be achieved by accelerating the craft on a track using linear motors, as implemented in maglev trains on earth. There are many implementations for transportation on earth, but so far this has not been used to propel a vehicle to lunar escape velocity. Such a launch system has been proposed for use on earth, where air resistance and a much higher escape velocity pose challenges not encountered on the moon.

Summary

Solar panels could be used to power propulsion systems which can be run on electricity. Ion drives provide such a propulsion system for use in space, and mass drivers provide such a system for launch.

Straying slightly from currently available technologies to those that are possible, but not yet achieved...

The Lunar surface is rich in Helium-3, so if Helium-3 fusion propulsion is developed, there is abundant fuel for it.

https://www.esa.int/Our_Activities/Preparing_for_the_Future/Space_for_Earth/Energy/Helium-3_mining_on_the_lunar_surface

While not really a propellant, since we as a civilisation started to explore the seas using sails, there's no reason why we wouldn't (at least at first) explore space in a similar fashion.

Enter Solar Sails:https://en.wikipedia.org/wiki/Solar_sail

Another idea would be to turn the moon in a 'laser beam hedgehog' and use powerful lasers to propel spaceships across our solar system, in a similar fashion to a solar sail.

Otherwise, at current, the best conventional engine is still the Hydrogen Engine: https://www.nasa.gov/topics/technology/hydrogen/hydrogen_fuel_of_choice.html
It (liquid hydrogen), however, would have to be obtained from water from ice mined from the asteroid belt and shipped to the moon.

Magnetic propulsion. For 6 days every month.

https://www.nasa.gov/images/content/222898main_orbit2_20080416_HI.jpg

The moon moves through the Earth's magnetic field in the course of its orbit. Once in the field, moon-based spacecraft could move via electromagnetic propulsion. This is not science fiction.

https://en.wikipedia.org/wiki/Electrodynamic_tether

Electrodynamic tethers (EDTs) are long conducting wires, such as one
deployed from a tether satellite, which can operate on electromagnetic
principles as generators, by converting their kinetic energy to
electrical energy, or as motors, converting electrical energy to
kinetic energy.1 Electric potential is generated across a conductive
tether by its motion through a planet's magnetic field.

Spacecraft with batteries (charged by solar panels during the other 24 days of the month) charge up their long tethers and use them to propel themselves about, pushing against the Earths field during its monthly visit.

Longer and more energetic tethers might be used all month long, pushing against the relatively weaker electromagnetic field of the sun and the charged particles of the solar wind.

Sodium in nuclear thermal rockets, or the same sodium as remass in an aluminium oxygen hybrid rocket.

Sodium with an atomic mass of 23, comes surprisingly good as an ntr propellant. Working near the melting point of uranium dioxide, the said propellant produced just over 300 seconds of ISP. Equivalent to that of modern storable propellants in term of specific impulse.

If combined with a graphite moderator, the ISP reaches 340 to 350s， high enough for most transfers, and certainly enough to reach nearby planets.

The sodium can be obtained as a byproduct from processing lunar feldspar rocks for aluminum construction material and breathing oxygen.

Sodium in a reaction of aluminum and oxygen greatly increases the thermal property of the plume and produces an ISP rating about 360 to 370s. This is viable for a Jupiter transfer.

Also, you can harvest lithium from lunar rocks and burn it in liquid oxygen. This affords more than 550 seconds of ISP, which is enough to send an interplanetary invasion. Lithium is also good for propellant of (artillery) guns, and as an energetic material in explosives.

Lastly, sodium and lithium can be used as a propellant for electric drives, which can be powered by silicon solar panels manufacturable from moon rocks. Or a fission reactor have to be shipped from the earth.

However, sending just the uranium 233 will not be too expensive, at least not as expensive as sending all the propellants up to the moon. This makes the whole fuel production process economically viable.

There is thorium ore on the moon, so nuclear power will be the go. Build the reactor on the moon, and you will be able to sell space service even back to the earth!

In summary, use alkaline metals to replace the hydrogen, and it shouldn't be too hard to build spaceships on the moon.

I strongly recommend two books that came out in the '70s

G. Harry Stine "The Third Industrial Revolution"

and

Heppenheimer's "Colonies in Space"

I see both in used book stores on a regular basis.

Stines notions was that you build orbital power satellites that would beam microwave energy down to receiving antenna on Earth. You use phased array to keep the beam narrow. No, the energy isn't enough to cook you if you are on the receiving antenna.

Material is mined on the moon and launched to a Lagrange point by a rail gun. There it is broken down using kilometer diameter solar mirrors. Much of the waste is silica -- which can be foamed and used as structural infill. Aluminum is the main structural material. Some silica is broken down to silicon (solar cells) and Oxygen (breathing) Hydrogen is in short supply. But if you can make oxygen, then you only have to ship up 1/8 the amount of rocket fuel you did before. And maybe those polar craters on the moon do have water in them.

Stine is convincing. He has an engineering background and had access to various think tank reports from the likes of the Rand Corporation.

Colonies in space is a bit further out and is more about establishing more than a work camp in zero-G.

Iron seems to be in short supply on the moon. H. proposes a nuclear rocket. Build a nuclear reactor that gets hot enough to turn gravel into hot gas. You can move asteroids then by landing such a rocket and a gravel crusher. At this point, I don't think we can make a nuclear engine that directly operates at those temperatures. Make electricity, make a plasma. Electrically accelerate the plasma. You can get huge specific impulse this way. It's not hard to get plasma up to a respectable fraction of light speed. It's more efficient however to accelerate more mass to a lower speed.

Be sure of your trajectories. Don't want to drop a 3-mile rock on the Earth by mistake.