Hi Kerrie,
The first measurements of the Sun's rotation were made by observing
sunspots with a telescope. From what I've read the first
documentation of this was probably by Thomas Harriot in 1610. The
first published observations appeared in 1611 by Johannes Fabricius who
observed spots for a few months. Christopher Scheiner actually
measured the rotation rate of the Sun at the equator and closer to the
poles of the Sun, noting that they were different in 1630.
Best,
Christina