You should feel good about being confused at this point, because it means that you're thinking deeply and trying to gain intuition about one of the most important parts of differential geometry — namely, how to compare vectors (and more generally tensors) at different points of a manifold. But there is no single answer to your questions, precisely because this ambiguity of how to "connect" the tangent space at one point to the tangent space at another is always present.
There are three main interesting approaches to the problem: the covariant derivative (or more general connection), the Lie derivative, and the exterior derivative. Even within those three, there are many choices to be made, rather than any canonically obvious choice for the "correct" derivative.
The one you use will be dictated by some other feature of your problem. For example, in physics, we usually assume that a metric exists with some meaning independent of coordinates. That, and a few more assumptions, are enough to basically specify one particular covariant derivative. Or there may be some special vector field (perhaps describing a physical symmetry), which will pick out a Lie derivative to use. Or maybe you only need to use differential forms (usually if you're integrating over the manifold or a surface in it), in which case you'll want to use the exterior derivative.
Now, as to parallel transport, this is defined in terms of a connection. In general, a connection is just exactly a function that tells you how much a given vector field changes as you move in a given direction. This is exactly the question you are pondering: how do you decide what the relationship is between vectors in two different tangent spaces? The connection tells you exactly how.
But connections are pretty arbitrary. They have to be linear, and they have to satisfy the Leibniz rule, but otherwise a general connection has basically no restrictions; you can make up your own. A vector field is said to be "parallel transported" along some curve if the field's covariant derivative along that curve is zero — which depends on your choice of connection. In fact, you could even just set up a frame (a set of vectors that span the tangent space) at each point, and define your connection so that they don't change from point to point. (At least, for some neighborhood of a given point.) Frankel points out that the natural connection for surveyors is exactly of this type, though it is not "metric compatible" and actually has torsion (and he references this paper for more details).
Having said that, if you have a metric (which we usually do in physics), then there is one common choice for the connection: the Levi-Civita connection — not the only choice, but a common one. It's special because it it has no torsion, and the metric is constant with respect to this derivative. It also has the nice property that the dot product between two vectors that are parallel-transported with respect to this connection is preserved.
So, the moral of the story is: parallel transport means more-or-less whatever you want it to. You decide what it means by choosing a connection. Therefore, if you want intuition about parallel transport, work on your intuition about connections.
