Does having vertex transitivity make the problem of calculating independence and chromatic numbers easier?



Vertex transitive graphs
Not particularly. There is a paper by Codenotti, Gerace, Vigna "Hardness results and spectral techniques for combinatorial problems on circulant graphs" Linear Algebra Appl. 285 (1998) 123-142 which shows that computing the chromatic number of a circulant graph is NP-hard.
(The pdf is available on Codenotti's web page.) Being vertex transitive guarantees that 
a $k$-regular graph has vertex connectivity at at least $2(k+1)/3$ and that its edge
connectivity is equal to $k$. Aside from this, it is not easy to identify useful consequences of vertex transitivity.


