For most of his first term, Washington managed to rise above partisan rancor. Democrat-Republican attacks on his administration (frequently the result of Jefferson's instigation), were generally focused on Alexander Hamilton, treasury secretary and committed Federalist. During his second term, this special status slowly vanished and he began to experience personal, and often brutal attacks by Democrat-Republican aligned newspapers. The most frequent critic of the administration was the Aurora, run by Benjamin Franklin's grandson. These attacks hardened Washington's positions, and while he still publicly opposed political parties, he privately identified with the Federalists and disparaged the Democratic-Republicans. (as an interesting side note, this stance alienated Washington from many of his peers in the Virginia gentry, who tended to be staunch Democratic-Republicans). The most common of these attacks was that he was a weak willed leader, who had been duped by Hamilton into enacting Federalist policies.

As to his retirement from the presidency, his private correspondence makes it clear that he was exhausted with the job and longed for retirement. He could have been elected to a third term had he wanted one, as he remained incredibly popular throughout the country. Many who advocated for him to remain viewed him as the only figure who could hold the new nation together, and feared the worst if he were to leave.

Finally, Washington sought to prove for most of his life that he did not seek power, he merely had it thrust upon him by circumstance. In leaving both the army and later the presidency, he sought to emulate Cinncinatus, the Roman hero who was granted total power, then surrendered it and returned home to his farm.

Source: Washington, a Life by Ron Chernow. Probably the best regarded Washington biography. It does a great job of disentangling Washington the man from Washington the legend.

As a follow up, what was public opinion like? I imagine he was well liked, but was he generally seen as an effective leader, or is it only in hindsight that we see how good of a politician he actually was?

Follow up: I've read that Washington was revered as almost God. So much so that there was once a golden statue of him positioned and looking much like Zeus. Was this something that began before, during, or after his terms as president? How much did he encourage it and if it wasn't from him, who started this? Also, where did it go and why? Seems to be a very dramatic rise and then revision.