I'm not very clever on mathematics and I'm not being able to start building a solution to this. I'm trying to develop an algorithm with which I can solve the following problem. Let's say I have a set of number $\{1, 2, 3, 4\}$, I want to go from $x$ ($x$ being in the subset $\{1,2,3\}$) to $4$ using all possible routes given that the possible routes are:
starting from $1$ $$ \{1,4\}, \{(1,2),(2,4)\},\{(1,3),(3,4)\},\{(1,2),(2,3),(3,4)\} $$ starting from $2$ $$ \{2,4\}, \{(2,3),(3,4)\} $$ starting from $3$ $$ \{3,4\} $$
As I said I'm not being able to develop a proper solution without doing it by hand, I would like to build an algorithm in which the input would be the boundaries of the set (in the case I presented $1$ and $4$) and the output would be the subsets with all possible combinations. Where can I start? Is this traveling salesman problem? What literature could I use to find out about this problem?
Unless I misunderstand your question, I think you missed e.g. $1\to3\to2\to4$.
You are using here permutation groups of the set $\{1,2,3,4\}$ and that would be a good thing to search for in the literature. These are subgroups of the symmetric group.
Note that this is often not always a commutative or abelian group because it matters in which orders you permute elements.
The transpositions are the elements of the permutation group that exchange two elements and leave all the others fixed. E.g. $(1,2)$ is the transposition that exchanges $1$ for $2$. You are constructing a chain of transpositions that exchanges $4$ with any element in $\{1,2,3\}$
A simple algorithm for your purposes begins with the observation that there will never be more than three steps (assuming you do not want to retrace your steps. If you did, there would be infinitely many solutions). Knowing this you can construct an exhaustive list of every chain of transpositions of length three. Then you can check each in turn to determine whether it satisfies your required conditions.
All your examples are increasing, but you don't give this as a rule. If you require your sequences to be increasing then you can just use this rule to filter out appropriate solutions.