Aristotle used the following four arguments for a spherical Earth (of which James K already discussed three):
During a Lunar eclipse, Earth's shadow is always round. If Earth had any other than round, you would sometimes see another shape.
The position of the stars are clearly different when observed in the north and in the south. For instance, he noted that some stars seen from Egypt and Cypres weren't visible from Greece at all.
In ancient times, there were elephants in North Africa, at least near the Strait of Gibraltar. If Earth were round, this would explain the existence of elephants there and in India, but not in Europe: The could simply walk around the other side.
One of the counterarguments against a round Earth was that people on the other side would fall off. Aristotle made this argument not a problem, but a solution: Massive things have a tendency to fall toward the center of the spherical cosmos, and if particles come from all sides toward one point, the resulting mass must look the same from all sides, i.e. spherical.
Argument #3 is… well… not the best argument. Argument #1 could in principle — and is in fact today by modern flat-earthers — be explained by lunar eclipses not being caused by Earth, but by a mysterious invisible "shadow object". Argument #2 can also be explained by the stars being much closer to Earth than we think. Finally, argument #4 is an fascinating prediction of how gravity really works, but at Aristotle's time it was mostly speculation.
Arguments that Aristotle did not use
Interestingly, despite being available to him there were several arguments that Aristotle didn't use:
One of the most common arguments for the sphericity of Earth's is that fact that you see the mast of an arriving ship before you see the hull. However, in certain ancient cosmological models, the Earth was not completely flat, but slightly curved like a shield, which could produce the same effect. Perhaps that is why Aristotle chose to disregard this argument.
Phenomena such as the sunset happen at different times, depending on where on Earth you are. Travelers from the north reported that the day was shorter up there, and the days passed faster when traveling from west to east than when traveling from east to west$^\dagger$. It is difficult to explain on a flat earth why the Sun cannot be seen by everyone at the same time, unless one simply assumes that the Sun shines like a kind of flashlight with a cone of light that only hits a limited area on the Earth's surface. This explanation was used, for example, in the flat-earth Gai tian cosmology of the 1st century BC. (but which is based on the roughly 1000-year-old world view Zhou Bi Suan Jing). It is also used by modern flat-earthers.
But the most important argument, which Aristotle did not make use of, is probably that when sailors crossed the equator, they saw the night sky revolve around a different star than what they were used to. To my knowledge this is not possible to explain on a flat Earth (but of course requires you to perform a journey, which isn't just something you can do while having a discussion with a flat-earther).
Today, we of course have many more ways of realizing that the Earth is round. However, as I described in an answer on physics.SE, none of these would enable you to convince a flat-earther since any argument that you come up with ultimately relies on your faith in previous people's observations and experiments (unless you want to derive all of physics and carry out all experiments in front of the flat-earther), and can thus be rejected either as "wrong" or "fake" or "how do you know that refraction works that way? Ah, you read it in a book?".
As others have noted here, the experiment carried out by Eratosthenes assumed, not proved, a round Earth.
$\dagger$In the encyclopedia Naturalis Historia from the 1st century AD the Roman historian Pliny the Elder describes how a messenger who ran the same route east and west noticed that the trip took longer from west to east, even though it was downhill.