It is true that every algebraic set is a finite union of algebraic varieties (irreducible algebraic sets), and this union is unique up to reordering. These irreducible pieces of an algebraic set are called the irreducible components. This all follows from the fact that a polynomial ring over a field is Noetherian, so that an algebraic set with the Zariski topology is a Noetherian topological space.
As an example, I always think of the algebraic set defined by the ideal $(xz,yz),$ which is not prime. The real picture of this algebraic set is a line through a plane, and these two objects are exactly the irreducible components of the algebraic set. Here is the picture:
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The components are defined by the prime ideals $(z)$ and $(x,y)$ which are the two minimal prime ideals containing $(xz,yz)$. This may be the eyeball test you desire, as most people would look at this set and say it is made of two parts. In general, the irreducible components of an algebraic set defined by an ideal $I$ correspond exactly to the minimal prime ideals containing $I$.
Concerning your second question, it is not easy in general to determine when an ideal is prime. I asked a question here seeking different techniques to detect when ideals are prime. It is often easier to see that an ideal is not prime, as in the example I've given.
I am not completely sure that this constitutes a satisfactory reply, but my answer is sheaf theory. Let me motivate this from the theory of manifolds.
If we have a manifold $M$, we can associate to any open subset $U \subseteq M$ the ring $\mathcal C^\infty(U)$ of $\mathcal C^\infty$ functions on $U$. It turns out that this makes $\mathcal C^\infty$ into a sheaf on $M$ (this means what I just said, plus some 'glueing' conditions that allow us to define such functions by defining them locally: instead of writing down a function on $U$, we may write down functions on open sets $U_i$ covering $U$, and if they agree on the intersections $U_i \cap U_j$, then they give a function defined on $U$).
Other important objects in the manifold world include the tangent space $T_p M$, the $i$-forms $\Omega^i(M)$, and maybe things like tensors (e.g. curvature is a $(0,4)$-tensor, meaning that it takes $4$ tangent vectors, and spits out a real number (which we think about as $0$ tangent vectors)). It turns out that all of these carry a natural multiplication structure by $\mathcal C^\infty$ functions. If we associate to each open $U \subseteq M$ the sets $\operatorname{VF}(U)$ (vector fields), $\Omega^i(U)$, etc, then this does not just become a sheaf, but it becomes a sheaf of $\mathcal C^\infty$-modules: over every open, we get a module over $\mathcal C^\infty (U)$, with natural compatibilities.
In algebraic geometry, it is not so clear what the tangent space or the $k$-forms should be: we certainly cannot use any differentiable structure to think about these (e.g. the tangent space is defined by infinitesimal paths through a point). It is here that modules step in: we can define a module of differentials $\Omega_{A/k}$ for any $k$-algebra $A$, which plays the role of $\Omega^1(M)$. Taking alternating powers (as an $A$-module!) gives the $\Omega^i_{A/k}$, and taking duals (again over $A$) gives $T_{A/k}$.
In the algebraic world, we can associate to any module $M$ over $A$ a sheaf $\tilde M$ over $\operatorname{Spec} A$, and this association is functorial in both $A$ and $M$. Moreover, the localisation $M_f$ corresponds to restriction to the open subset where $f$ is nonzero. We can now glue the modules $\Omega_{A/k}$ to define a sheaf of differentials $\Omega_{X/k}$ for any variety $X$ over $k$. Moreover, analogously to the analytic case, this is a sheaf of $\mathcal O_X$-modules.
So one could ask: why not stick to vector bundles? In the algebraic world, these correspond to finite projective modules (or equivalently, finite flat modules). However, we quickly run into problems if we do that: for example, the cokernel of a map of projective modules need not be projective (just think about $\mathbb Z \stackrel{n}{\to} \mathbb Z$ as $\mathbb Z$-modules). So if we want to apply any sort of homological methods (which prove to be very powerful in algebraic geometry), we better have a theory that works at least for all finitely generated modules. This gives the notion of coherent sheaves.
Observe, however, that the notion of a coherent sheaf is not restricted to algebraic geometry; if we have any ringed space $(X, \mathcal O_X)$ (both varieties and manifolds are examples of this), then we can define what it means for a sheaf of $\mathcal O_X$-modules to be coherent. It is a theorem that this notion corresponds to the notion of finitely generated modules when we are working with a variety.
On the other hand, for smooth projective varieties over $\mathbb C$, Serre's GAGA paper proves that the categories of coherent modules in the algebraic and the analytic world are equivalent! Thus, our notion is truly very close to that of (holomorphic) vector bundles over a complex manifold.
As we showed in the example above, there are many natural examples of vector bundles over manifolds, and we can now define all those things on varieties as well. But there are many more uses for modules: for example, line bundles correspond to divisors (codimension $1$ subvarieties) modulo some equivalence. Ample line bundles classify morphisms into $\mathbb P^n$. So there are lots of different examples where a priori algebraic objects (modules/coherent sheaves) turn out to have important geometric interpretations.
Best Answer
In short: "varieties are nice schemes over fields". In particular, the study of varieties is a subscase of the study of schemes. A variety (depending on who you ask) over a field $k$ is a finite type separated (usually reduced, sometimes geometrically integral!) scheme over $\text{spec}(k)$.
The reason that the two fields look so different, is purely in language. When one studies varieties on their own, it's likely that one is using a source that uses the classical (read "pre-Grothendieck") language. This language suffices to talk about such well-behaved schemes, but fails to distinguish between the more sophisticated properties of general schemes (e.g. a point $\text{spec}(k)$ and a 'fuzzy point' $\text{spec}(k[x]/(x^m))$).
That said, much of general scheme theory actually can be reduced to the study of varieties. This is because, for a sufficiently nice schemes (most that we encounter) we can think about them as being 'fibered' over varieties. By this, I mean that if $X$ is a scheme, with an equipped map of schemes $X\to S$ which satisfies some mild properties, then all of the fibers $X_s$ for $s\in S$ are varieties over $k(s)$. This allows one to think about general schemes as being 'families of varieties indexed by other schemes'.
It is a mistake to confuse 'variety land' with 'commutative algebra land'. What is more likely happening, is that when you are looking at texts on varieties, they happen to be focusing (perhaps at the beginning) on affine varieties. This is why the following correct identification can be confused with the above incorrect one: 'affine scheme land' IS 'commutative algebra land'. This holds true for schemes over arbitrary bases, and is made precise by an (anti)equivalence of categories between affine schemes (over an affine base) and algebras over that affine base.
In terms of learning it, historically varieties came before fields. They are also the most tenable examples of schemes (besides, perhaps, number rings), and so are useful to have in your back pocket. The classical language, what you would most likely learn varieties in, has the advantage of being simpler, and perhaps easier to see the geometry. Unfortunately, it is often times sloppy, and hard to analyze the fine structure results that you'd need.
In this way, I think that a cursory reading of a book on varieties, and almost more helpful a book on complex manifolds, is a helpful first step to studying schemes. That said, there will be no technical loss (only a loss of intuition) by starting directly with schemes.
Good luck, and feel free to ask a follow up question.