Translating a vector field along the x-axis? Say you've been given 6 points which represent a vector in a vector field: 
 $ x, y, z, u, v, w $. 
$ x, y, z$ will be pointing somewhere from the origin (0, 0, 0) to a point in space. From this point in space the vector (which will be shown in the vector field) is represented from that point in space to the point $u, v, w $. This represents the vector in its vector field.
Now for my question: how does one go about translating this vector field to say (1, 0, 0)? That means to say that the $ x, y, z$ vector starts at (1, 0, 0) to the point in space. And the vector represented by (u, v, w) should also now be in the offsetted position?
Am I confused to believe that this is something non-trivial?
 A: Short answer: no, you are correct in believing that this is non-trivial.
More detail/pointers: a vector field in a space is really a choice, for each point $p$ in the space, of a vector in a vector space attached to that point, say $V_p$. If I understand your question correctly, $(x,y,z)$ would be coordinates of the point $p$ and $(u,v,w)$ would be coordinates for a vector in $V_p$. 
Crucially, there is not, in general, any way to naturally identify vector spaces $V_p$ and $V_q$ when $p \neq q$ are different points in space (and I have been deliberately vague about what the "space" might be).
The proper context for the question, in this generality, is differential geometry, specifically vector bundles and connections on them. Briefly and roughly, the vector bundle contains all possible vector fields and a connection is a way to move a vector from one $V_p$ to another. The result will in general depend on the path chosen, which is captured by the notion of holonomy. 
It is not possible to give any useful summary of these matters in a post like this, but I'd like to say some words about why this machinery can sometimes be avoided. Doing so requires me to be explicit about what "space" means.
If by "space" we mean "smooth manifold", we are in the general situation sketched above, and there are many different connections.
If by "space" we mean "Riemannian manifold", there is a special connection, the metric connection, which is derived from the metric. 
If by "space" we mean Euclidean space, then we have a Riemannian manifold which is flat (has zero curvature), in which case the path-dependence mentioned above does not occur. For this reason, it is possible to completely omit mention of connections and holonomy, and pretend that all different vector spaces $V_p$ are the same. This is why it is possible to talk about vector fields without all the machinery of differential geometry. However, you incur a sort of "technical debt" (figuratively speaking), which must be repaid if you ever want to move on to more general contexts.
I should add that there are other situations than Euclidean space where these questions can be simplified/elided, but I am not trying to achieve any completeness here, merely give illustrations/context. 
In summary, what seems like a simple question is really the tip of an iceberg. How you want to go about learning about this depends, of course, on your background. At a minimum, you'll have to decide what kind of space you are working with, since the amount of machinery needed depends drastically on what structure your space affords.
