Ok, let's start from the beginning, and you tell us if any of these steps doesn't work:
1. On your phone, open the Google Play Store, and select the Apps store.
2. Select an app that you want to install. If it isn't already installed on your phone, you should see an "Install" button on the app page (if it's free, otherwise there will be a purchase button).
3. If you press Install, it will show you the permissions that the app needs--if you agree, then press Continue.
4. Your phone should tell you when the app has finished installing.
5. On the app page in the Google Play Store, you should now see an "Open" button and an "Uninstall" button.
6. If you exit out of the Play Store and go to your Apps Drawer, you should see an icon for the app you just installed.
Are there any steps here that aren't working for you?