The paper deals with the ground and the first excited state of the polaron in the one-dimensional Holstein model. Various variational methods are used to investigate both the weak-coupling and strong-coupling case, as well as the crossover regime between them. Two of the methods, presented here, introduce interesting elements to the understanding of the nature of the polaron. Reliable numerical evidence is found that, in the strong-coupling regime, the ground and the first excited state of the self-trapped polaron are well described within the adiabatic limit. The lattice vibration modes associated with the self-trapped polarons are analyzed in detail, and the frequency softening of the vibration mode at the central site of the small polaron is estimated. It is shown that the first excited state of the system in the strong-coupling regime corresponds to the excitation of the soft phonon mode within the polaron. In the crossover regime, the ground and the first excited state of the system can be approximated by the anticrossing of the self-trapped and the delocalized polaron state. In this way, the connection between the behavior of the ground and the first excited state is qualitatively explained.