The main idea of this paper is to propose a nondestructive evaluation (NDE) system for two types of damages, core cracking and skin/core debonding, in fiberglass/foam core sandwich structures based on the inverse eigensensitivity-based finite element model updating using the modal test results, and the artificial neural networks. First, the modal testing was conducted on the fabricated fiberglass/foam core sandwich specimens, in the intact and damaged states, and the natural frequencies were extracted. Finite element modeling and inverse eigensensitivity-based model updating of the intact and damaged sandwich structures were conducted and the parameters of the models were identified. Afterward, the updated finite element models were employed to generate a large dataset of the first five harmonic frequencies of the damaged sandwich structures with different damage sizes and locations. This dataset was adopted to train the machine learning models for detection, localization, and size estimation of the core cracking and skin/core debonding damages. A multilayer perceptron neural network classification model was used for detection of types of damages and also a multilayer perceptron neural network regression model was fitted to the dataset for automatically estimation of the locations and dimensions of damages. This intelligent system of damage quantification was also used to make predictions on real damaged specimens not seen by the system. The results indicated that the extracted natural frequencies from the accurate finite element model, in coordination with the experimental data, and using the artificial neural networks can provide an effective system for nondestructive evaluation of foam core sandwich structures.