I heard that Newton used the Moon acceleration to come up with his universal law of gravity, but in order to calculate its acceleration you need the distance to the Moon. Whose measurement of this distance was used and how did they measure it?
In Newton's time distance to the moon was not a problem: since the ancient times this distance (in terms of the Earth radius) was measured using parallax. Newton used the value 60 for the ratio of Moon's orbit to the Earth radius. This is close to the value 59 which can be found in Ptolemy.
Newton's problem was with Earth radius which was not known to young Newton with sufficient accuracy. For this reason his first check of the gravitation law gave incorrect result, and Newton abandoned the subject. Only when a more precise measurement became available (by Picard), Newton saw that his gravitation theory is correct.
Reference. The article Newton's moon test by Douglas W. MacDougal in the book Newton's gravity, Springer 2012, gives the following values of the distance available in Newton's time: 59 (Ptolemy), 60 (Huygens), 60 1/3 (Copernicus), 60 2/5 (Street) and 60 1/2 (Tyho Brahe). Newton does not mention all these authors, just taking 60 as a "well-known" number.
Remarks. Since the question of size the the Earth is raised in the comments, let me comment on this too. The best measurement which Newton could use is by Snell (6369 km), but Newton did not know this result at the time of calculation.
Wikipedia claims that Biruni's measurement was accurate to 17 km (0.2%), which is hard to believe, given his method described in Wikipedia. I followed the links from Wikipedia article, and could not trace the reference for this unbelievably precise number. Quick search hows various figures attributed to al-Biruni, and the ridiculously precise value cited in Wikipedia is certainly made up.