Numerical ratio between the length of a string and the transited states If we have a finite automaton that accepts a string,

is there any numerical relationship that always is true like S ≥ |w| or something similar? 

Where |w| the length of the string and S the number of transited states (counting multiplicities).

For example, if we have this automata and the string w = 010 :



S = 3 and |w| = 3. 

In this case, S = |w|

I'm new in computer science and i was thinking there has to be a rule of that kind but I have not found explicitly in the books that I started on computational theory.
The way you have defined S, we will always have $S \ge |w|$.  Why?  Because each time the automaton reads one symbol from $w$, it follows one transition and visits the state at the end of that transition, incrementing $S$ by one.  In other words, we add one to $S$ for each symbol in $w$.  Thus $S$ is at least as large to the number of symbols in $w$.  (It might be larger if you follow $\epsilon$-transitions, as they don't consume any symbols in $w$ but do visit a state.)

Try working through a few examples and it should become clearer.