I saw several answers here close to what I wanted to say, but none of them quite do. So:
Plastic Wrap Not Needed
If you supply the Moon with an atmosphere, that atmosphere will not be instantly lost to space. On human time scales the Moon holds most atmospheric gases quite well.
However, it will slowly bleed water away to space. So the important question is, "how long can the Moon hold onto its atmospheric water?" According to my calculations (table below), atmospheric water has a half-life of almost 400,000 years.
Half-life of gases on significant bodies:
Key:
- Black letters, red background - gas half-life < $1 \cdot 10^8$ years.
- Orange letters, yellow background - $1 \cdot 10^8$ years < gas half-life
< $4 \cdot 10^9$ years.
- Green letters, green background - $4 \cdot 10^9$ years < gas half-life.
- Blue lettered "liquid" - substance is a liquid at these conditions.
- Green lettered "solid" - substance is a solid at these conditions.
- White letters, brown background - two phase solid/gas $4 \cdot 10^9$ years < gas half-life.
- Yellow letters, brown background - two phase solid/gas $1 \cdot 10^8$ years < gas half-life < $4 \cdot 10^9$ years.
- Red letters, brown background - two phase solid/gas gas half-life < $1 \cdot 10^8$ years.
- White letters, blue background - two phase liquid/gas $4 \cdot 10^9$ years < gas half-life.
- Yellow letters, blue background - two phase liquid/gas $1 \cdot 10^8$ years < gas half-life < $4 \cdot 10^9$ years.
- Red letters, blue background - two phase liquid/gas with gas half-life < $1 \cdot 10^8$ years.
NOTE: I consider the substance to be two phase when its partial pressure at these conditions exceeds 0.01 bar (1% of Earth's atmospheric pressure). Anything below that and the loss of gases to space will be significantly reduced due to the small amount of the substance in the atmosphere. Also because it was exceedingly difficult to find the conditions required for partial pressures lower than 0.01 bar for many of these substances.
So for a long-lived technologically sophisticated civilization, it may just be easier to periodically refresh the Moon's surface water inventory than to build and maintain a plastic wrapping.
How to accomplish it anyway
Assuming you want to throw caution to the wind and construct the Moon's plastic wrapper anyway...
Materials
Start with composite materials.
For the matrix material, you will want a durable plastic that's not reactive to either the conditions in space or the atmosphere. The plastic must also be transparent. Something like this PVC (polyvinyl chloride) that's been mixed with a number of other plastics to make it inert.
Then embed either glass (for transparency) or carbon (for strength) fibers. You'll want to lay-up the fibers at 60 degree intervals to accommodate stresses in all directions.
Structure
Rather than making one continuous sheet of plastic across the Moon. Make this as a bunch of interconnected smaller domes. Select a standard dome size and make this the size of each segment.
Each dome will consist of a minimum of two layers of dome material. Separate the layers by about 32 feet. Fill the inter layer region with pure water. This provides your terraformed habitat with 3 essential things:
- Radiation shielding.
- Mass to counter (some of) the pressure of the atmosphere
- A thermal reservoir to help mitigate some of the extreme temperature
swings from the 28 day long light/dark cycle.
Shape the structure so that the carbon fiber reinforcing strands reach down to anchor points in the Moons surface. The carbon fiber strands anchored to the surface of the Moon provide the rest of the necessary force to hold the gases in.
Each of the dome's sides will include something like "tent flaps". This allows a dome segment to retain atmosphere when an undomed (or depressurized dome) section is adjacent to it. But the flap will be raised when there is an adjacent dome and it is pressurized. Ideally the raising and lowering of flaps will occur automatically.
Domes will include anchor & sealing points on their exterior surface. This will allow a "dome" patch to extend from the 6 adjacent domes and cover a dome that has been damaged or needs maintenance.
Construction
To construction this in sections, make each segment a hexagonal dome. As you add segments, attach them to the appropriate face the adjacent dome. When first added, the plastic layer will extend all the way down to the surface.
Making a tight seal with the surface of the Moon will be difficult. It probably requires a special construction effort to create a rim made of some concrete or plastic analog at the dome edges. This rim probably needs to extend down into the Lunar bedrock and be formed of an inert material too.
The designers & builders would embed airlocks in the dome rim structure so they wouldn't have to put holes in the dome structural materials. Each connection face would likely have at least one airlock connection.
Timing
This is a massive effort. Before you finish the project, the original domes will probably require complete refurbishment. Some method of sealing domes away from the vacuum of space and each other must be included in the structure. That enables the construction crew to repair dome segments without total depressurization.
After completing a dome segment, the terraforming crew will pump the proper hydrospheric and atmospheric substances into the segment. Expect the Moon's surface to react, possibly violently, with the materials for a while. I presume that additional oxygen and water will need to be pumped back in after a time since the initial mix will likely react with the Lunar surface.
Including "port holes" in the structure
Simply leave some hexagonal segments empty. Then the adjoining segments would provide the airlock facilities required to access the area of vacuum.
Similarly, industrial process which could take advantage of the vacuum of space, could leave adjoining segments empty too. However, if they're exhausting corrosive substances, the builders will want to coat the dome materials with extra substances to make it inert to the caustic materials being exhausted at that point.
Living in this biosphere
Unfortunately, we really don't have any data for how things will do in this environment. Therefore, the biologists in charge of the effort might keep several of the domed hexagons sealed off from each other and experiment with different biospheres in each segment. Once the find some biospheres that work, then they might simply open the "tent flaps" and allow that one to spread across the surface.
Of course the main problem will be the day-night cycles. This will be extremely difficult for the plant life to live with. Therefore, the domes will likely possess an artificial light source to provide light during the long lunar night.
The temperature extremes will also pose a problem. The water trapped between the two dome layers will help but won't solve the problem. Including large bodies of water (water filled craters, anyone?) will help moderate temperatures but probably the Lunar surface will require additional heating and cooling. I expect the dome structures as I've envisioned them to interfere with natural atmospheric convection that might otherwise help moderate those temperatures. However, the surface could have built in heat pumps. During the long day, they exchange heat between the surface and the bottom of nearby crater-lakes - cooling the surface. During the long night, they do the same in reverse - heating the surface.
Fluffy Lunar Atmosphere
Because the Moon's gravity is so much less than Earth's, the Moon's atmosphere will be much deeper. You will also need 6x the mass of atmosphere in a column to make the same pressure that you see on Earth. What this means is that the Moon requires nearly the same mass of gases as the Earth does for a given pressure.
- Earth's atmosphere mass = $5.1 \cdot 10^{18}$ kg
- Moon's atmosphere mass (for 1 bar pressure) = $2.3 \cdot 10^{18}$
In fact, the Moon's atmospheric profile ought to look very similar to Titan's (at least the pressure would - probably need to add ~200 K to the temperature though):
Titan Atmosphere Profile