Absorption, fluorescence, and magic-angle pump-probe experiments characterize the solvatochromism and relaxation dynamics of three structurally related near-infrared tricarbocyanine dyes (HDITCP, IR125, and IR144) in solution. Agreement with solvatochromic theory is found in solvents where the conductivity approximately matches that predicted for complete ionic dissociation. The nonpolar solvatochromism of HDITCP and IR125 allows the polar solvatochromism of the IR144 absorption spectrum to be attributed to a specific functional group. Resonance structure arguments predict a dipole moment decrease upon electronic absorption by IR144, consistent with the observed solvatochromism. Assuming a point dipole, spherical cavity reaction field model, self-consistent feedback between the solvent and the polarizable IR144 solute accounts for 1/2 to 1/3 of the observed polar solvent shifts. A geometry change in the excited state leads to nearly nonpolar solvatochromism in the IR144 emission spectrum. Femtosecond magic-angle pump-probe transients show similar underdamped intramolecular vibrational quantum beats in all three molecules, but find solvent-dependent overdamped responses on a picosecond time scale in all three dyes. The quantum beat decays determine inhomogeneous vibrational dephasing times of a few picoseconds. Vibrational relaxation, nonpolar solvation, and dielectric relaxation all take place on similar time scales, but the picosecond relaxations are all slower and of larger amplitude for IR144 (polar solvatochromism) than HDITCP (nonpolar solvatochromism). In contrast to HDITCP, IR144 has a prominent solvent-dependent "coherence spike" near T = 0 which is attributed to femtosecond polar solvation dynamics.