As @MSalters says the key measure is delta-V, the total amount by which the spaceship can change the velocity of its payload.
Starting from the ground you need about 10 km/s to reach a stable orbit. From there you need about 5.5 km/s to land softly on the Moon and about another 2.5 km/s to get from the Moon's surface back to Earth where you can use the atmosphere to slow down.
How much delta-V a rocket gives you depends on the fuel used and the engine efficiency, which together give a number called the specific impulse $I_{sp}$ and the mass ratio, the ratio of the dry mass (engines, empty tanks, cargo, crew, etc.) to the fuelled mass (all the above plus fuel and oxidiser).
The Starship concept comes with a first stage booster called Superheavy, which has not yet been built. Current estimates (from wikipedia) put its dry mass at 230 tons and its wet mass at 3530 tons for a mass ratio of about 14.7. The $I_{sp}$ of the Raptor engine is about 330s.
So if you had no upper stage at all, you can calculate the delta-V of Superheavy as about 8.8 km/s. Not quite enough to reach orbit, although it's close. So no matter how small you make the second stage it's not going to be able to launch it to the Moon. The reason is that you are lifting the huge superheavy tanks and engines to orbit, which is wasteful.
If you add an upper stage, you get less delta-V from Superheavy, because you have increased the dry mass, but you get to add the delta-V from the upper stage. So Starship has a wet mass of about 1320 tons and a dry mass (no cargo or crew) of about 120 tons for a Mass ratio of 11 (less than SuperHeavy because it needs a crew cabin and stuff and because bigger tanks naturally have less surface area compared to their volume). The $I_{sp}$ is a bit higher because the engines work better in vacuum, so you get about 8.8 km/s of delta-V from Starship (with no cargo).
If you put a fuelled starship on a superheavy, the "dry" mass (when the Superheavy is out of fuel) is now about 1550 tons, and the fuelled mass about 4650 for a mass ratio of 3, and we get about 3.5 km/s of delta-V from Superheavy, so ignoring many details, we make orbit with a fuel reserve enough for about 2.3 km/s. Not quite enough to get to the Moon, let alone land.
We could try different upper stages. The Centaur, for example, has a wet mass of about 23 tons (there are multiple variants) and a dry mass of 2.2 tons, with an $I_{sp}$ of 453s (using liquid hydrogen fuel). This gives a delta-V of just over 10 km/s with no payload. With only this payload superheavy gets about 8.5 km/s. So this configuration has, in principle, enough delta-V to get to the Moon' surface and back, probably with enough left over for a few hundred kg of payload.
This is, however, a fantasy experiment, for a whole bunch of reasons. First of all, the centaur upper stage fuel won't remain liquid for long enough to get to the moon and back. Adding active cooling would add a lot of mass. Secondly landing on the Moon would need legs and other added mass. Thirdly the upper stage could be quite a lot bigger and still not reduce the delta-V of Superheavy that much.
At the end of the day, though, the question is "why?". If spaceX can make refueling work it will be a cheap and effective solution to the problem. If not, the obvious way to the Moon is more stages (Apollo style).
