Verification will depend on Coinbase's policies, and may vary depending on how much money you are are dealing with.
You haven't yet done anything that touches the Ethereum network. You've opened an account at Coinbase. Coinbase have lots of Ether. When you bought them, they credited some of them to you in their internal database.
When you want to actually use your Ether to send transactions or buy something from someone, you'll need to get them out of Coinbase. The normal thing to do is to create your own wallet, and withdraw the Ether. When you do this you get an address from your wallet, and tell Coinbase that address, and debit your account in their database and they send you some of the Ether from their stash by making a transaction on the Ethereum network.
If you're just sending payments rather than using DApps then it may be possible to just leave the Ether with Coinbase, and "withdraw" directly to the address that you're trying to pay. But it's usually better to have your own wallet, make a single once-off withdrawal and interact with the Ethereum network yourself, rather than having Coinbase do it on your behalf all the time.