A semiflexible Gaussian chain model is used to determine the statistics and correlations of single-molecule fluorescence resonant energy transfer (FRET) experiments on biological polymers. The model incorporates a persistence length in a Rouse chain and describes single-chain dynamics with normal modes. The hydrodynamic interaction is included in the dynamics of the semiflexible Gaussian chain on the preaveraging level. The distribution functions of the fluorescence lifetime and the FRET efficiency provide direct measures of the chain stiffness, and their correlation functions probe the intrachain dynamics at the single-molecule level. When measured with finite time resolution, the instantaneous diffusion coefficient for FRET is much smaller in the collapsed structure than in the coiled structure, and the variation has a quadratic dependence on the donor-acceptor distance. In the fast reaction limit, single-molecule FRET lifetime measurements can be used to map out the equilibrium distribution function of interfluorophore distance. As an example of microrheology, the intrinsic viscoelasticity can be extracted from single-molecule tracking of the Brownian dynamics of polymers in solution.