Do I have to look at the screen all the time?

Personal experience:

The screen needs to be on and Pokemon Go needs to be active.
It will vibrate when a Pokemon is near.

When you get the accessory called Pokemon Go Plus it will notify when Pokemon are near even when the screen is not active.

Pokemon Go has a power save mode, when you turn it on it can be turned up side down in your pocket, the screen will go blackish with the Pokemon Go logo on it, it will vibrate if a Pokemon is near

In order to receive notifications about nearby Pokemon and have steps tracked for egg counting, you must have the screen on and the app in the foreground.

You may use Battery Saver Mode to darken your screen and avoid looking at your phone. When the phone is turned upside-down (in your pocket, for example), the screen will darken but the app will continue running in the foreground.