Revisit the definition of a vector space: a set of vectors that is closed under vector addition and scalar multiplication.
To extrapolate: suppose we have a set of vectors in $R^n$, called "S." Vectors in this set are all kinds of vectors with variables $x_1, x_2,...,x_n,$ as long as for any two vectors $u,v \in S$, $\space \space \space u+v\in S$ as well, and for any scalar $k\in R$, $\space \space k\cdot u \in S$ also.
You could give set S any quality you like- that doesn't necessarily mean it is a vector space. Take, for example, the set of all vectors $(x_1, x_2,1)$. If I had two vectors in that form, they do not form a vector space- after all, here is a counter example: $(3,2,1)$ and $(5,4,1)$. Add them together and you get $(8,6,2)$, which doesn't have a "1" in the right-most entry.
So no, you don't just add two vectors to create a vector space (clearly, the one above fails). A vector space is defined by the constraint you put on it, as long as that constraint keeps the space closed under vector addition and scalar multiplication.
Vector spaces are useful for determining the visual representations of solutions to linear systems, and are a fundamental part of understanding not only linear algebra, but higher math topics (i.e., commutative algebra) as well. Best to get comfortable with them as soon as you can!