Intersection yes - you can formally prove it by checking that the intersection set of V,W satisfies all the axioms of vector space

Union no - as a counter example x-axis and y-axis are subspaces of R^2. yet the union of two i.e. co-ordinate axis is not a sub-space. infact it should be good to try what further condition should be imposed on V,W so that the union is a vector space. it is pretty trivial