I've been reading that Minkowski space is a real vector space; atleast in flat space-time. So that means that we should be able to define vector addition, scalar multiplication, etc. But since four-vectors in space-time are events, how does adding two events make sense? More importantly, what is the meaning of the resultant vector?
Minkowski spacetime is an affine space not a vector space: vectors acts as group of translations on the spacetime moving events, but vectors are not events. If you arbitraily fix an origin $O$, there is exactly one vector $v(e)$ joining $O$ to every event $e$ of the spacetime. In this sense there is a one-to-one correspondence between events and four-vectors, but it strongly depends on the choice of the preferred origin.
If you also fix a pseudo-orthogonal basis, the components of $v(e)$ with respect to that basis are just the Minkowskian coordinates of $e$.