Is there a way to generalize the connected sets in $\mathbb{R}^2$? How do I generalize the connected sets in $\mathbb{R}^2$ if there is a way. I know how connected sets in $\mathbb{R}$ look like, so is there a way to say something about the connected sets in $\mathbb{R}^2$?
 A: There is no good way to classify the connected sets in $\mathbb{R}^2$. Even the planar continua - that is, connected subsets of $\mathbb{R}^2$ which are closed and bounded - are extremely varied (and have a very interesting theory!), see e.g. here.
Now granted, since there isn't a good rigorous notion of "classification" it's hard to turn this into a precise theorem. However, I think the following pretty is pretty definitive:

$(*)\quad$ There are $2^{2^{\aleph_0}}$-many homeomorphism types of connected subsets of the plane. In fact, this is true even for simply connected and path connected subsets!

That is, there are as many distinct "topological types" of connected sets in the plane as there are sets in the plane in the first place! Contrast this with the situation for $\mathbb{R}$, where there are only four.

The proof of $(*)$ is a bit lengthy, but ultimately elementary. Here's one way to do this:
Basically, we want to associate to each set $A$ of paths through the infinite binary tree (there are $2^{2^{\aleph_0}}$-many sets of such paths) an $X_A\subseteq\mathbb{R}^2$ which is connected and such that $A\not=B$ implies $X_A\not\cong X_B$. The first step to doing this is to build a copy of the complete binary tree inside $\mathbb{R}^2$, in a "topologically-decodable" way. For example: 
Here the root of the tree is recognizable as the only degree-five point, and we can distinguish left and right via the degree-four points (the little hash marks in the middle of the left-pointing edges). Of course I've only drawn the first few branchings of the tree.
Now we attach, to the "end" of each path through this infinite tree, a copy of $[0,1]$ or $[0,1)$ depending on whether the infinite binary sequence representing that path is in $A$
This is all a little vague; it's a good exercise to fill in the details. Note that the sets so constructed are topologically quite nice in may ways (e.g. contractible, and locally homeomorphic to $\mathbb{R}$ except on a nowhere-dense set).
