Use labels and filters in your inbox.
Labels and filters are also a "0 inbox" solution that will help you not miss any important emails and stop postponing your answers by getting only what concerns you.
A filter is a rule that GMail will apply when you receive an email that match.
A label is a sort of a "folder" for your emails. You can also have sub-labels.
Use the "archive" and "apply a label" options together
With this solution, you can read what you receive in your priority inbox because it really concerns you and requires your attention, and if you want, you can read your other emails later by browsing your labels. Also note that it is possible to configure notifications on the Android GMail app based on the labels. It's probably also available on the iOs app.
