For an affine scheme $S=\operatorname{Spec}(A)$ the following are equivalent:
1) $A$ is zero-dimensional
2) $S$ has all its points closed (i.e. $S$ is $T_1$)
3) $S$ is Hausdorff
4) $S$ is compact Hausdorff
If moreover the ring $A$ is noetherian the above are also equivalent to:
5) $A$ is Artinian
6) $S$ has the discrete topology
7) $S$ is finite and has the discrete topology
If moreover the ring $A$ is finitely generated over a field $k$ (and thus noetherian) the above are also equivalent to:
8) $A$ is algebraic over $k$
9) $ \text {dim}_k(A)\lt \infty$
10) $\text {Card}(S)\lt \infty$
Examples of non-noetherian rings satisfying 1) to 4):
Any infinite product of any fields $A=\Pi_{i\in I} K_i$ ($I$ infinite)
Examples of noetherian rings which are not algebras over a field but satisfy 1) to 7):
$\mathbb Z/(n)$ with $n\gt 1$ and not prime.
Edit. It was already asked before on math.SE how convergence looks like in the Zariski topology.
There are lots of connections between topological properties of $\mathrm{Spec}(R)$ and algebraic properties of $R$. They can be found in many introductions to algebraic geometry or commutative algebra, for example Atiyah's book, Eisenbud's book, EGA, etc. But the topology doesn't recover the whole algebraic structure, since for example the spectrum of a local artinian ring consists just of one single point; more generally $\mathrm{Spec}(R)$ is homeomorphic to $\mathrm{Spec}(R_{\mathrm{red}})$. This means that the topology can only "see" the ring modulo nilpotents. This is why affine schemes come equipped with the structure sheaf, and are able to reconstruct the ring completely. Anyway, here is a list of the connections:
$\mathrm{Spec}(R)$ is always spectral, i.e. it is sober, quasi-compact, and has a basis consisting of quasi-compact open subsets which is stable under intersections.
$\mathrm{Spec}(R)$ is Hausdorff iff $\mathrm{Spec}(R)$ is $T_1$ iff $\mathrm{dim}(R)=0$ (R. Gilmer, Background and preliminaries on zero-dimensional rings, Zero-dimensional Comutative Rings, Marcel Dekker, New York, 1995, pp. 1-13).
$\mathrm{Spec}(R)$ is connected iff $0,1$ are the only idempotents of $R$. More generally, if $X$ is a locally ringed space, then $f \mapsto X_f$ is a bijection between idempotents in $\Gamma(X,\mathcal{O}_X)$ and clopen subsets of $X$.
There is an anti-isomorphism between the closed subsets of $\mathrm{Spec}(R)$ and the radical ideals of $R$. It follows, for example, that $\mathrm{Spec}(R)$ is noetherian as a topological space iff every ascending chain of radical ideals of $R$ becomes stationary. For example, $\mathrm{Spec}(R)$ is noetherian if $R$ is noetherian (but not conversely).
Under the anti-isomorphism above, the irreducible closed subsets correspond to the prime ideals of $R$. It follows that the topological dimension of $\mathrm{Spec}(R)$ coincides with the Krull dimension of $R$.
The closed points of $\mathrm{Spec}(R)$ correspond to the maximal ideals of $R$. For example, $\mathrm{Spec}(R)$ has a unique closed point iff $R$ is local.
The generic points of $\mathrm{Spec}(R)$ correspond to the minimal prime ideals of $R$. It follows that $\mathrm{Spec}(R)$ is irreducible iff $R$ has only one minimal prime ideal iff $\mathrm{rad}(R)$ is a prime ideal.
Additional remarks:
(a) Not every connected affine scheme is path-connected, see here.
(b) More sophisticated topological properties can be found at the Stacks Project, see topology for the notions of catenary, jacobson, constructible, proper, etc. and algebra for the corresponding properties of rings.
(c) Convergent nets or sequence aren't used so often in algebraic geometry. Even the definition of a complete variety (more generally proper morphisms) doesn't refer to the usual notion of completeness in topological spaces, but rather to a more abstract one where the net is replaced by a $K$-valued point, and its limit is replaced by an $A$-valued point, where $A$ is a valuation ring and $K$ is its field of fractions. Similarily, algebraic geometry needs a more abstract notion of the Hausdorff separation property, namely that of separatedness, which can be characterized by the condition that the abstract limits above are unique iff they exist. In general, one has to adjust the notions and techniques from topology and differential geometry in order to apply them to algebraic geometry. Sometimes even the topology has to be adjusted, for example the Inverse Function Theorem becomes only valid in the etale topology.
Of course you can try to find convergent nets or sequences in (affine) schemes, but in my opinion this is not the best way to get comfortable with them, and probably nets are not really useful in this context, since open sets tend to be quite large. For example, an algebraic curve carries the cofinite topology, plus a generic point, which implies that every sequence of pairwise distinct elements converges to any point.
Best Answer
This is the constructible topology, the topology generated by both the Zariski open sets and the basic Zariski closed sets (i.e., the closed sets defined by principal ideals). Another way to say it is that it is the topology generated by declaring the distinguished Zariski open subsets $D(f)$ to be clopen, rather than just open. Or, it is what you get by considering $\operatorname{Spec} R$ as a subset of $\mathcal{P}(X)\cong \{0,1\}^X$ and giving it the product topology, where you put the discrete topology on $\{0,1\}$. (This description is immediate from your description of what limits look like, since you are just taking pointwise limits in $\{0,1\}^X$.)
It can also be described purely topologically from the Zariski topology: it is the topology generated the sets that are compact and open in the Zariski topology, together with their complements. This topology is useful in the abstract theory of spectral spaces (spaces that are homeomorphic to the spectrum of a ring with the Zariski topology); see https://stacks.math.columbia.edu/tag/08YF for instance. It is also of interest because the constructible sets (in the sense of algebraic geometry) are exactly the clopen subsets in the constructible topology (so the constructible topology makes $\operatorname{Spec} R$ the Stone space of the Boolean algebra of constructible sets).