For your doorbell, you can get Notifications for both your Motion in front of your doorbell and also for when someone pushes the doorbell button.
Main Menu (the 3 horizontal line icon, upper-left-corner) > Devices > select your doorbell > then toggle ON (swipe rightward) for your "Ring Alerts" and "Motion Alerts."
If these already are toggled ON while you are not receiving Notification, then you should 'resend' your setting to the Ring server cloud by cycling your toggles OFF and then back ON to ensure your correct desired settings are stored correctly on your Ring account.
Also insure your cellphone settings do not have notifications for the Ring App turned off, preventing the App's notifications from getting displayed.
Additionally, there are other Ring App settings, such as "People Only Mode", "Motion Verification", or "Motion Frequency", and "Advanced Motion Detection" that can impact/reduce some of the Notifications that think you should be receiving. So you should also double check those settings.
Although you only asked about those text-type Notifications for your doorbell, remember you can also get App Alert Tones sounds for you to hear.
From the Doorbell page, select the "Device Settings (the blue Gear icon) > Alert Settings > App Alert Tones > choose your desired sounds for both "Rings" and for "Motions."