Directional, differential and lie derivatives on manifolds intuition? Trying to translate elementary multivariable calculus into the language of manifolds:
Is the directional derivative on a manifold just a way of finding the rate of change of a vector in a single direction, in a fixed basis, where the vector is expressed in terms of linear combinations of basis vectors (i.e. to find the rate of change of a vector in the tangent space moving along one of the directions the basis points out)?
Is the differential then just a way of finding the rate of change of that same vector in a single direction, where the vector is just expressed in terms of it's coordinates in an invariant fashion (i.e. to find the rate of change of an equivalent vector in the cotangent space that is dual to the original vector in one direction directed out by the basis)?
If I want to take a second derivative in another direction given by a fixed basis, am I forced to define the lie derivative of a one-form? i.e. is the lie derivative just a fancy way of taking second derivatives of scalar-valued functions in a single direction (while also interpretable as first, second, ... derivatives of vector-valued functions)?
In terms of vector-valued functions of vector fields, I've never understood why the second derivative naturally ends up with us having to define a bilinear form, intuitively why does this necessarily arise in taking the second derivative of something like $\vec{F}(x,y) = (x^2+y^2,2xy)$?
How does this intuitive example translate into the language of lie derivatives of vector fields?
Is the covariant derivative just a way to do all of the above in an arbitrary basis, i.e. in a random direction no matter what basis we're given?
What does the commutator actually do, thinking along these lines?
 A: This doesn't answer all of your questions, and some of the answers may be lower-level than what you hope, but I detect what I think is some confusion in your question, and I am going to try to explain things from first principles. I've been waiting for someone to answer this question, and I haven't seen anyone try.
I'm not clear which bilinear forms you're talking about, so I'll remain silent on that.
The way I think of it, directional derivatives aren't "rates of change" of vectors, but are identified with vectors, and are applied to functions from the manifold to the reals. That is, I think of a local direction as a directional derivative in that direction.
Take a chart on a manifold and use it to pretend the domain of that chart is an open set in Euclidean space. 
http://en.wikipedia.org/wiki/Tangent_space#Tangent_vectors_as_directional_derivatives
The total derivative of a map from one Euclidean space to another is the linear map that, at a point, best approximates that map. If one takes the directional derivatives with respect to the standard basis (the partials) as a basis for the tangent space, then the Jacobian matrix of partials of the coordinates of that function represents the total derivative. To apply this idea to a map $f\colon N \to M$ of manifolds, compose with charts on either side. To apply the differential $f_*$ to a tangent vector $X \in T_p N$, remembering that a tangent vector is thought of as a derivation on the algebra of germs of functions at a point, one just precomposes it:
$$(f_* X)\colon \phi \mapsto X(\phi \circ f)$$
for a germ $\phi$ of functions at $f(p)$ in $M$.
The thing to understand about this is that if we ignore the charts and say we're working with maps between Euclidean spaces, and write $X$ in terms of a basis of partial derivatives, then $f_*$ is represented by the Jacobian matrix $[\partial f^i/\partial x^j]$.
The Lie derivative on vectors is a way of differentiating vector fields, essentially: $\mathcal L_X \tilde Y$ is the rate of change of the vector field $\tilde Y$ in the $X$-direction, where $X \in T_p M$. The problem is in saying what "the $X$-direction" is. If a manifold is equipped with a Riemannian metric, there's a map called the exponential from a neighborhood of the origin in the tangent space to a neighborhood of $p$ that says what it means to "keep going in the $X$-direction" beyond $p$.
The problem is that the "first directional derivatives" are all maps between different tangent spaces, and unlike in $\mathbb R^n$, there's no canonical identification of tangent spaces, so one has to compare things in different tangent spaces to take the second derivatives. The point of a connection is to that let you do that.
