Data integration broadly defined as making use of data from multiple sources is not a single thing. The "making use of" part determines the kind of integration. One example of data integration in when you map various pieces of information to a common reference genome, another example is statistical data integration to combine data for analysis. The approach you take is determined by the context and the question you try to answer. Nevertheless, a common first step is to define the data sources to use. Should you include data from study A but not from study B? When using curated resources, you may need to be aware that they contain redundant information or biases that may affect your conclusions.
Regarding the more specific question of integrating public databases into a single repository, the question would be: for what purpose and in which form? Consider that we already have plenty of such integrated databases such as Ensembl or all the model organisms databases (e.g. wormbase, flybase...) each adapted to serve a specific purpose or community.