Blame me, my parents, or my schools as you see fit, but after half a century as an American citizen, 13 years of public education, and a college degree, I couldn't name all of the presidents of the United States, much less in order. The mystery is why no one ever tried to teach me, given how easily I learned them when I put my mind to it, and how handy it has been (and would have been in history class!) to have even a rough idea of who fits in where.
Actually, I did not even have to put my mind to the problem, only my ears. I bought a copy of Sue Dickson's "Song of the U.S. Presidents," and after a few hearings it stuck. It's not a great song, but as with many not-so-great songs, that seems to make it stick all the better. (The link takes you to an updated version that I haven't tried yet (mine ends with Clinton), but the sample suggests it is basically the same.) Of all the U.S. President songs, that one is my favorite, because it is short, simple, and easy to rattle off mentally when needed—such as when I'm playing the "put the pictures of the presidents in chronological order" game with my nephews. However, it teaches only the order (no numbers) and gives last names but not first, so you have to know which Adams is which, and which Harrison, and that both Clevelands are the same person.
There are other songs designed to teach the presidents, most of which I find more obnoxious than helpful, but I recently discovered one I like a lot for the other information it presents. Along with each president's name (sometimes full, sometimes unfortunately only the surname), you hear his number and a clue to something special about him or that happened during his presidency. For this reason, it's not as good for quick recall as Dickson's song, but I expect it to be a valuable tool for sorting and organizing our country's history.
The song was written by Genevieve Madeline Ryan in response to a challenge by her father to memorize the presidents for Father's Day. You can see a Flash presentation with pictures at the White House Historical Association website ("The American Presidents," about 2/3 of the way down on the right-hand side of the page). You can also obtain a version for yourself. I found their system for downloading somewhat confusing (the "Download" button actually takes me to a podcast feed), but you can download the video, audio, and lyrics by right clicking on the links I've provided and saving the files to your computer.