The Skinny: Wikipedia has the following detailed plot summary for the book:
Gonsales is a Spaniard forced to flee the country after killing a man in a duel. Having made his fortune in the East Indies he decides to return to Spain, but falls ill on the voyage home and is set off on the island of St Helena to recover. There he discovers a species of wild swan able to carry substantial loads, the gansa, and contrives a device that allows him to harness many of them together and fly around the island. Once fully recovered Gonsales resumes his journey home, but his ship is attacked by an English fleet off the coast of Tenerife. He uses his flying machine to escape to the shore, but once safely landed he is approached by hostile natives and is forced to take off again. This time his birds fly higher and higher, towards the Moon, which they reach after a journey of twelve days. There Gonsales encounters the Lunars, a tall Christian people inhabiting what appears to be a utopian paradise. After six months of living among them Gonsales becomes homesick and concerned for the condition of his birds, and sets off to return to Earth. He lands in China, where he is immediately arrested as a magician, but after learning the language manages to win the trust of the local mandarin. The story ends with Gonsales meeting a group of Jesuit missionaries, who arrange to have a written account of his adventures sent back to Spain.
Of interest, this book draws upon theories presented by Kepler and Galileo, so it is more than just a work of fantastical fiction as it tries to incorporate scientific elements known at the time. This book was preceded by Kepler's own Somnium which is also considered an important early example of science fiction. You can read more about The Man in the Moone at its Wikipedia entry.
Where Can I Find It? Interestingly, this book does not appear to be on the Project Gutenberg site which has many, many public domain works, but you can find a pdf version of it at this link. You can also get a free audiobook version of the work over at LibriVox. I'm guessing this one might be a bit of a tough read (or listen) because of its age, but may still be worth checking out for curiosity's sake.