I start with Newton's method for the square root of two or three. This also leads to a simple answer that I don't need a debugger to check. I start with a simple loop, then move that to a subroutine, then I move the subroutine to a differnt file so I can learn to make a library. Then I expand the function in the library to handle different root finding.
in reply to What do you write when learning a new language?
I did this in Pugs to try out Perl 6 and liked it so much I wrote an article about it. But, I've done it for many other languages over the years including Perl 5 and even (shudder) COBOL.