I just compiled bitcoind and run for the first time. Do I already have a private key? Since I have a wallet.dat I suspect I already have a private key and so I could extract it for backup purpose (on paper). 

How do I do it via terminal with bitcoin-cli?
Bitcoin Core will automatically create a new wallet on first start. This wallet will contain 2000 private keys (unless otherwise specified by the -keypool=&lt;n&gt; option) and the seed used to generate those and future keys, at least until you encrypt the wallet.

As such, there is not just one private key that you need to backup, rather you need to backup 2000 keys and the seed. Otherwise, restoring a backup will not restore all of your private keys.

The easiest way to backup your wallet is to just copy the wallet.dat file to a safe place. I would not recommend that you try to create a paper backup of your wallet.

If you insist on a paper backup, then you should record the seed. The seed can only be gotten by using the dumpwallet RPC command and then finding the private key labeled hdseed. The seed is a private key and cannot be used in any other wallet software since they do not accept the seed as a private key. You will need to remember that it specifically is not just a normal private key and that it only works in Bitcoin Core. To restore the seed, you can use the sethdseed command in a new wallet.

I really don't recommend that you try to backup the seed. There are changes happening to the Bitcoin Core wallet soon that may make your seed backup unable to work. The recommended way of backing up is to copy the wallet.dat file, and to do so periodically. This will guarantee that you have your private keys, guarantee that it is compatible with future versions of Core, and also backup transactions and metadata (such as labels).