Hybrid search is one of the features that tries to blur the lines between your on-premises and SharePoint Online environments. As you begin to migrate more data into the cloud, the users will require hybrid search functionality to quickly and easily find the files they require. The setup process is relatively long and complicated, and there are plenty of things that can and probably will go wrong. The best advice I can give is to test as much as possible at every stage and ensure you don't try to skip any steps or assume that services have been set up correctly in the past. There are six steps that we will go through in our lab environment and as usual I encourage you to do the same in yours before attempting any configuration changes in your production environment.
Let's dive in and look at the steps involved, and look at the potential issues to watch out for:
Configure AD sync
Configure STS trust
Configure server-to-server authentication
Configure result sources...