(From Vance Randolph, Ozark Folksongs, volume IV, p. 407, as supplied by Pearl Craig)
George Washington, first president,
By Adams was succeeded,
And Thomas Jefferson was next,
For the people’s cause he pleaded.
Madison was then called forth
To give John Bull* a peeling,
James Monroe had all the go,
The era of good feeling.
’Twas J. Q. Adams then came in,
An’ next came Andrew Jackson
Who licked John Bull at New Orleans
With such great satisfaction.
Van Buren then he took the chair,
Then Harrison and Tyler,
The latter made the Whigs so mad†
They like to bust their b’iler.
We then elected James K. Polk,
The issue that did vex us
Was shall we fight with Mexico
An’ take in Lone Star Texas?
Taylor then got in the chair,
But soon had to forsake it,
Millard Fillmore filled it more,
Frank Pierce then says I’ll take it.
James Buchanan next popped in,
Lincoln then was chosen,
Who found the events of his might (sic.)
Were anything but frozen.
Andrew Johnson had a time,
The Senate would impeach him,
But as it took a two-thirds vote,
They lacked one vote to reach him.
Now we come to U. S. Grant
Who made his name at Shiloh,
Then Hayes, an’ Garfield who was shot,
They both came from Ohio.
[Arthur] then the scepter held
Till Cleveland turned it over,
Little Bennie sandwiched in,
An’ now again it’s Grover.
* John Bull: Great Britain. A reference to the War of 1812.
† John Tyler was William Henry Harrison’s Vice President, and succeeded when Harrison died in 1841. Harrison was a Whig, but Tyler aligned himself with the Democrats.
§ Little Bennie=Benjamin Harrison, who lost the popular vote to Grover Cleveland in 1888 but won the Electoral College. Cleveland won their rematch in 1892.
The Minnesota Heritage Songbook 
The version I learned in school bad different. It started:
George Washington, the first of all,
By Adams was succeeded;
And then came Thomas Jefferson, Who bought the land we needed.
I would like to find that version again.
This is a song that many teachers modified for themselves, e.g. adding new presidents as they were inaugurated. The best place to start would likely be the school where you learned it. This version does not sound familiar to me — but it might well be a local variant. That line about Jefferson buying “the land we needed” sounds very manifest-destiny-ish; it might have been changed after manifest destiny went out of fashion, or it might have been added by some particularly pro-expansion teacher.
That’s the version I learned next line is
John Quincy Adams was the next
And then came Andrew Jackson
And after him Van Buren came
With panic and distraction
That’s all I can remember