I'm going to go a bit overboard here and give you the sketch of how vectors are geometrically constructed, since I think it's helpful to know. While writing this I found I was phrasing things very carefully, which means: you may need to reread parts of this in a quiet corner if it doesn't all make sense at first.
Suppose that you have a set of scalar fields $\mathbf{S} \subseteq (\mathcal{M} \rightarrow \mathbb R) $ on an underlying set of "points" $\mathcal{M}$. This set S contains every scalar field that you are interested in. It's helpful to define some jargon: let an $n$-functor be a smooth functions in $\text C^\infty(\mathbb R^n \rightarrow \mathbb R)$. We can of course apply them to real values (which I'll denote with parentheses as $\Phi(\dots)$, but also we can use them to combine scalar fields, which I'll denote with square brackets as $\Phi[a, b, \dots]$ which is the function from any $p \in \mathcal M$ to $\Phi(a(p), b(p), \dots)$. Henceforth I will just call scalar fields scalars, since we're talking about field theory.
When we close S under $n$-functors for all $n$, we get a lot of stuff: first off we can add and multiply scalars since +
and *
are 2-functors; but we can even define a topology on $\mathcal M$ with those scalars which makes all of them continuous (the kernel topology; $s \subseteq \mathcal M$ is closed when $s = \operatorname{ker} \sigma$ for some $\sigma \in \mathbf S$), and we can use this topology to define whether $\mathcal M$ is connected or not, etc. If you go down this rabbit hole far enough, you get to specify criteria by which $\mathcal M$ is a "manifold". But let's get more concrete and specific.
Now in special relativity in particular, there are special fields $w, x, y, z$ which we use as global coordinates, meaning that every scalar $\phi$ in $\mathbf{S}$ can be written as a 4-functor $\Phi(x_1, x_2, x_3, x_4)$ applied to those scalar fields; $\phi = \Phi[w, x, y, z].$ (In general relativity, you still have coordinates, but you will have to use the topology to define "neighborhoods" of a point and then say that for every point there is a neighborhood enclosing that point with a set of $d$ coordinates which distinguish points in that neighborhood.)
We can take your intuitive 3D notion of "vector fields" and associate them with derivations $D_v = v^\mu \partial_\mu$, so that we can define vectors (again dropping the word "field") in a purely geometric sense as the set of linear maps $V \in (\mathbf S \rightarrow \mathbf S)$ satisfying the chain rule: given a $k$-functor $\Phi(x_1\dots x_k)$ with partial derivatives $\Phi_{(i)} = \partial \Phi/\partial x_i$, then $$ V [\Phi[s_1\dots s_k]] = \sum_{i=1}^k ~ V[s_i] ~ \Phi_{(i)}[s_1\dots s_k]. $$ We can then identify the components of any vector field by applying it to our coordinates, so you see there is a two-way correspondence between these abstract mathematical operations on S and the lists-of-coordinates that you're used to.
We can write this space of derivations on $\mathbf S$ as $\mathbf S^\bullet$, and make independent copies of it for every symbol that we want to stick up top: this is known as "abstract index notation". We write elements of this set as symbols with a corresponding superscript; so $v^\alpha \in \mathbf S^\alpha$.
Then there are covectors, like the general gradient of a field, which are linear maps from vectors to scalars. For example, given a scalar $\phi$ there is a covector $\nabla\phi$ defined as $$\nabla\phi~(V) = V[\phi].$$ These we can denote by $\mathbf S_\bullet$ and we copy the space for every symbol. Now you know what $\partial_\mu \phi$ means; it is the copy of $\nabla \phi$ that lives in the space $\mathbf S_\mu$. It is a covector field. (Somewhat important: you don't quite know what $\partial_\mu v^\nu$ means yet. More complicated derivatives of this kind -- derivatives of vectors rather than scalars -- require some added geometric structure called a "connection" on the manifold.)
Finally we have the metric, which is a linear map from a pair of vectors to a scalar; this can be used to convert covectors to vectors.