Do truly absolute directions exist? If not, what directions exist independent of coordinate systems? I feel that, until I understand the answers to these questions, I will not fully understand how vectors exist independent of coordinate systems.
No. If what you mean by "directions" is a set of independent (unit) vectors whose span is the vector space, then any invertible linear transformation will give you a different set of "directions."
To address your second point which is to understand what vectors are, it's best to begin with axiomatic vector space theory (c.f. Wikipedia). Vectors are elements of a vector space. Once you understand the properties of vector spaces (and their structure-preserving maps), you can understand that "directions" (more commonly called bases) are any linearly-independent spanning set on the vector space, and is moreover not unique (the sets are related by these maps). In fact for a vector space of dimension $n$, any set of $n$ linearly-independent vectors constitutes a basis. A coordinate system is then the choice of one such basis: an arbitrary vector in this vector space can then be expressed as a linear combination of such basis elements (by the spanning property), and the resulting coefficients of such a combination are the coordinates of the vector.
One example is the direction between two stars in a constellation. The relative position between then changes so little in a timelife that can be taken as fixed.
But for that ones like ursa major in the northern hemisphere or crux in the southern one, all the group rotates along one night. A direction between 2 stars parallel to the horizon at $9$PM becomes perpendicular six hours later.
Nothing moved with the vector, but our (informal perception) coordinate system has rotated.