The three-step two-color resonant ionization method and three-step three-color isolated-core excitation (ICE) technique are used to study the spectra of the highly excited bound states systematically, either Eu 4f76snl Rydberg states or other valence states converging to the higher ionization limits. Specifically, the highly excited bound states are populated from the ground state via three different 4f76s6p intermediate states, thereby establishing the three different excitation schemes. The schemes are designed to allow us to assign a J-quantum number uniquely to a given highly excited state with the selection rules of J-quantum number for each excitation scheme by comparing their corresponding photoionization spectra, which are obtained with three-step two-color resonant ionization method. By tuning the wavelength of the second laser, the 56 highly excited bound states located in the energy region between 42250 cm-1 and 44510 cm-1 are detected. To explore their spectroscopic information, more efforts have been made 1) to judge whether an excited state is a bound Rydberg state and to observe whether it may be excited further to an autoionizing state by using the ICE technique; 2) to deduce the principal quantum number of the given bound Rydberg states, and to observe whether they are converged to the same ionization limit by calculating their quantum defects with respect to several ionization limits. Based on the above manipulations, all detected highly excited bound states can be classified as the two categories: bound Eu 4f76snl Rydberg states and other valence states converging to the higher ionization limits, such as the Eu 4f75dnl states. Specifically, to fulfill the ICE technique, it is necessary to make a resonance transition from the 4f76snl Rydberg states to the 4f76p1/2nl autoionizing states with the third dye laser whose wavelength is scanned around the Eu 4f76s+-4f76p1/2+ ionic line. Once the Eu 4f76snl Rydberg states are recognized with the ICE technique, the identification of their orbital quantum numbers is a primary task to determine their electron configurations. With all the efforts mentioned and existing information, three Rydberg states can be assigned to the 4f76s10s(8S9/2), 4f76s9d(8D9/2) and 4f76s9d(6D7/2), whereas the rest can be regarded as highly excited valence states.