We discuss approximation algorithms for the coloring problem and the maximum independent set problem in 3-uniform hypergraphs. An algorithm for coloring 3-uniform 2-colorable hypergraphs in Õ(n1/5) colors is presented, improving previously known results. Also, for every fixed γ > 1/2, we describe an algorithm that, given a 3-uniform hypergraph H on n vertices with an independent set of size γn, finds an independent set of size Ω̃(min(n, n6γ-3)). For certain values of γ we are able to improve this using the local ratio approach. The results are obtained through semidefinite programming relaxations of these optimization problems.