Who first measured the distance to the Moon? How was it done? I think it had to happen after Newton, but I am not sure.
It happened long before Newton. In the second century BC Hipparchus used lunar parallax to calculate a value for the minimum and maximum distance of the earth and moon. His results are very close to the modern calculation of this distance.
You can read about it here: Toomer G.J. (1974), "Hipparchus on the Distances of the Sun and Moon." Archives for the History of the Exact Sciences 14: 126–142.
This is an ill defined question. It can be interpreted as "Who was the first to TRY to measure the distance to the Moon", or "Who was the first to give a correct number", and what is counted as correct number.
And even this does not define the question precisely. The crucial question: "in what units"? It is relatively easy to measure the distance in terms of the radius of the Earth. But it is another matter, if you want the answer in miles, kilometers or stadia.
The earliest written evidence that survived is a purely theoretical work of Aristarchus where he explains some mathematics involved. One can conclude from the book that he hardly measured anything in practice.
Hipparchus who lived later, had already the idea of the order of magnitude of Moon's parallax (this is equivalent to measuring the distance in terms of the Earth radius). However most ancients did not discuss the distance as we understand it, but only ratios, for example the ratio of the distance to the Sun and to the Moon, or the Moon's parallax which is sufficiently large to be measured by primitive tools. To pass from the Moon parallax to the distance in some conventional length units one needs to know the size of the Earth. The size of the Earth was measured by Eratosthenes, but it is still subject of discussion how accurate his measurement was.
The problem is that he gives the answer is stadia, and nobody knows how long his stadium was. Few other measurements were made after him, but still when Newton was a student he was taught that the degree of the meridian is 60 (British) miles. There was no way to measure the distance to the Moon without knowing the radius of the Earth.
Speaking of accuracy: Hellenistic Greeks could measure the Moon parallax to minutes (Ptolemy), and this was not improved until good instruments were manufactured at the time of Brahe. Since then accuracy constantly increased. Precise measurements of the size of the Earth are even later, they begin in the end of 17 century.