Your marginal distributions are correct. But I get E(X) = E(Y) = 2/5.

There are several options for finding the pdf of U = X + Y.

One option is to use the Change of Variable Theorem (see for example Walpole, Myers and Myers - Probability and Statistics For Engineers and Scientists): Let U = X + Y and V = X. Get the joint pdf of U and V and then calculate the marginal pdf for U. I get g(u) = 4u^3.

Another option (more work) would be to use the method of distribution functions (see for example Mendenhall, Scheaffer and Wackerley - Mathematical Statistics With Applications).