The goal of this paper is to propose fully discrete local discontinuous Galerkin (LDG) finite element methods for the Cahn-Hilliard-Navier-Stokes (CHNS) equation, which are shown to be unconditionally energy stable. In details, using the convex splitting principle, we first construct a first order scheme and a second order Crank-Nicolson scheme for time discretizations. The proposed schemes are shown to be unconditionally energy stable. Then, using the invariant energy quadratization (IEQ) approach, we develop a novel linear and decoupled first order scheme, which is easy to implement and energy stable. In addition, a semi-implicit spectral deferred correction (SDC) method combining with the first order convex splitting scheme is employed to improve the temporal accuracy. Due to the local properties of the LDG methods, the resulting algebraic equations at the implicit level is easy to implement and can be solved in an explicit way when it is coupled with iterative methods. In particular, we present efficient and practical multigrid solvers to solve the resulting algebraic equations, which have nearly optimal complexity. Numerical experiments of the accuracy and long time simulations are presented to illustrate the high order accuracy in both time and space, the capability and efficiency of the proposed methods.