The availability of patient cohorts with several types of omics data opens new perspectives for exploring the disease's underlying biological processes and developing predictive models. It also comes with new challenges in computational biology in terms of integrating high-dimensional and heterogeneous data in a fashion that captures the interrelationships between multiple genes and their functions. Deep learning methods offer promising perspectives for integrating multi-omics data. In this paper, we review the existing integration strategies based on autoencoders and propose a new customizable one whose principle relies on a two-phase approach. In the first phase, we adapt the training to each data source independently before learning cross-modality interactions in the second phase. By taking into account each source's singularity, we show that this approach succeeds at taking advantage of all the sources more efficiently than other strategies. Moreover, by adapting our architecture to the computation of Shapley additive explanations, our model can provide interpretable results in a multi-source setting. Using multiple omics sources from different TCGA cohorts, we demonstrate the performance of the proposed method for cancer on test cases for several tasks, such as the classification of tumor types and breast cancer subtypes, as well as survival outcome prediction. We show through our experiments the great performances of our architecture on seven different datasets with various sizes and provide some interpretations of the results obtained. Our code is available on (https://github.com/HakimBenkirane/CustOmics).