a little clarrification - you don't take the covariant derivative
w.r.t. a vector, but of a vector w.r.t. a neighbourhood. you do take
the Lie derivative w.r.t. a one parameter group of diffeo's and this
is related to the vector field. all this can be found in wald's book,
or nakahara as mentioned before.

a counter to the statemtent:

"Intuitively, it appears that the covariant derivative with respect to
a
particular vector V is just the Lie derivative taken along a geodesic
to
which V is tangent."

is simple one! consider the covariant derivative of the metric and the
Lie derivative of the metric and you get the point.

or, have a a look at equation 10.9.10 in weinberg's book and you'll
see that the Lie only reduces to the covariant in the special case
where certain terms vanish.

hope that helps.

paul