Alzheimer's Disease (AD) accounts for the majority of dementia, and Mild Cognitive Impairment (MCI) is the early stage of AD. Early and accurate diagnosis of dementia plays a vital role in more targeted treatments and effectively halting disease progression. However, the clinical diagnosis of dementia requires various examinations, which are expensive and require a high level of expertise from the doctor. In this paper, we proposed a classification method based on multi-modal data including Electroencephalogram (EEG), eye tracking and behavioral data for early diagnosis of AD and MCI. Paradigms with various task difficulties were used to identify different severity of dementia: eye movement task and resting-state EEG tasks were used to detect AD, while eye movement task and delayed match-to-sample task were used to detect MCI. Besides, the effects of different features were compared and suitable EEG channels were selected for the detection. Furthermore, we proposed a data augmentation method to enlarge the dataset, designed an extra ERPNet feature extract layer to extract multi-modal features and used domain-adversarial neural network to improve the performance of MCI diagnosis. We achieved an average accuracy of 88.81% for MCI diagnosis and 100% for AD diagnosis. The results of this paper suggest that our classification method can provide a feasible and affordable way to diagnose dementia.