In this paper, channel estimation techniques and phase shift design for intelligent reflecting surface (IRS)-empowered single-user multiple-input multiple-output (SU-MIMO) systems are proposed. Among four channel estimation techniques developed in the paper, the two novel ones, single-path approximated channel (SPAC) and selective emphasis on rank-one matrices (SEROM), have low training overhead to enable practical IRS-empowered SU-MIMO systems. SPAC is mainly based on parameter estimation by approximating IRS-related channels as dominant single-path channels. SEROM exploits IRS phase shifts as well as training signals for channel estimation and easily adjusts its training overhead. A closed-form solution for IRS phase shift design is also developed to maximize spectral efficiency where the solution only requires basic linear operations. Numerical results show that SPAC and SEROM combined with the proposed IRS phase shift design achieve high spectral efficiency even with low training overhead compared to existing methods.