Answer: Mr. Lincoln's first official act was to appoint John Nicolay as his secretary. Nicolay, at a salary of $2500, servedthroughout the Lincoln Administration, and along with co-worker John Hay, wrote an extensive Lincoln biography, the only one approved by Robert Todd Lincoln.
Source: David Herbert Donald's
We Are Lincoln Men, page 180. and
http://civilwarstudies.org/trivia.shtm#29
Edit - As I've said before, it is my practice to give credit for alternative answers if a source can be found to support them. In this case, I believe at least one source can be found to support every answer that has been given, so I will give credit for all answers.
Although the source originally presented in Faraway Friend's post does not link to the short item stating that the inaugural address was Lincoln's first official act, this link does:
http://www.ushistory.org/us/32d.asp
Brenal provided two books in support of his contention that provisioning Fort Sumter was Lincoln's first official act. I happen to have a copy of Sandburg's book, so I checked through it. The provisioning is, in fact, the first act that is mentioned in Sandburg's book, even though I couldn't find anyplace where Sandburg stated explicitly that it was Lincoln's first official act, so I will accept that answer as well.
But here's a surprise - while we are now well accustomed to the idea that it is standard procedure for a new president to give his inaugural address after being sworn in, according to Sandburg, that was not the case in 1861.
Chapter 4 of Sandburg's book states that Sen. Edward Baker (who would later be killed at Ball's Bluff) stepped forward to introduce Abraham Lincoln as "President-elect of the United States." Then came the inaugural address. And after that, Chief Justice Taney stepped forward and administered the oath of office.
So, technically, Lincoln was not yet President when he delivered his inaugural address, because he hadn't yet been sworn in.
Which means that Mark F. Jenkins' answer, that Lincoln's first act as President was to remove his hand from the Bible, might be the best answer of all.
Hoosier