If you’re having trouble with the Google Play Store app not downloading, please see the steps below.
Step 1: Check that you have a strong Wi-Fi or mobile data connection.
First, make sure you have a strong internet connection. It’s best if you can connect to a Wi-Fi network. If you don’t have access to Wi-Fi, make sure you have a strong mobile data connection. Then try your download again.
Get more help with connection problems on Android devices.
Step 2: Clear the cache & data of the Play Store
This gives the app a fresh start and can help fix issues.
1. Open your device’s Settings app Settings.
2. Tap Apps & notifications and then See all apps.
3. Scroll down and tap Google Play Store Google Play.
4. Tap Storage and then Clear Cache.
5. Next, tap Clear data.
6. Re-open the Play Store & try your download again.
Step 3: Restart your device
If you still can’t download after you clear the cache & data of the Play Store, restart your device.
1. Press and hold the Power button until the menu pops up.
2. Tap Power off or Restart if that is an option.
3. If needed, press and hold the Power button until your device turns on again.
If you’re still having trouble
If these steps don’t work, see other troubleshooting steps.
Hopefully useful.
Source : Google Play Help