By default, the YouTube app will play 480p when you first open a video. It is assumed to switch to your screen resolution a few seconds in, but this is not always the case. The usual YouTube app doesn't let you change this behavior, but like most of Android there is still a way.
There is a modified version of YouTube, called YouTube Vanced, which packs a number of features not found in the version available in the Play Store. One of these features includes changing the default quality for when you are on cellular data and Wi-Fi, so you don't have to wait for the app to switch over. While there is a method for rooted users, the process we discuss below does not require rooting.
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YouTube-vanced must be sideloaded, so make sure "Unknown Sources" or the "Install Unknown Apps" state is enabled on your phone. If you want to sign in to the app with your regular YouTube account to see your subscriptions and look at the history, you also need to download an add-on called MicroG. MicroG is an open source switch for Google Play Services, and although it may be difficult to install it, it is just as easy to sideline any other app.
To start, press the link below from your phone and select "MicroG Advanced (New)" to download the extension, then press "Install" when prompted. Then go back to the same page and select the first entry in the list to download YouTube Vanced. Once again, tap "Install" when prompted.
After installing the app, you have two YouTube apps on your phone, the standard YouTube app and YouTube Vanced. You can easily hide the other app using custom launcher like Nova.
Be warned that since you load this app page, there is a risk that the APK file might contain malicious code. Unlike the Play Store where the app is reviewed by Google, you trust the mud to deliver what they promise. This is a very popular app with a large group of users, so you should be good, but please download at your own risk.
Step 2: Change the default playback quality
Now open YouTube Vanced, and in the right corner, select your profile picture (or silhouette icon) and then select "Settings". Select "Advanced Settings" and then select "Video Settings".
Select "Preferred Video Quality Mobile" and select the desired screen resolution for videos you watch on when connected to mobile data. Then select "Preferred video quality Wi-Fi" to change the default resolution when using a Wi-Fi connection, and then select a resolution.
In addition to changing the default playback resolution, you can also let your device play back higher resolution video than yours screen. With this extra resolution, the video is reduced to the screen resolution, giving a sharper image thanks to additional data. But be aware that using a higher resolution consumes more data. Therefore, if you do not have an unlimited data plan, you can only standardize the higher image (such as 2160p or 4K) on the Wi-Fi connection.
Once selected, press your back button until you return to the main page and open a video. As long as the video producer uploads the video at the desired resolution (or higher), it should start with that resolution instead of 480p. If you have chosen a different Wi-Fi resolution, connect to a Wi-Fi network and watch another video.