Are you using mobile data or WiFi? How strong is your network signal? If mobile data isn't working, have you tried WiFi? If WiFi isn't working, have you rebooted your router?
Welcome to Android Central! Try going to Settings>Apps, select Google Play Store, tap Storage, and Clear Cache/Clear Data, then Force Stop. Do the same for Download Manager -- you probably have to tap Menu>Show System to see it. Now try again.