I see it lets me back up the wallet.dat, but don't I need the private keys?
No. The wallet is a collection of key-pairs
I obviously have my passphrase, but what about the private keys? Do I need to export them somewhere or am I ok just to click on "back up wallet" and save it in multiple spots?
You don't need them, and if you wallet is a HD wallet you technically don't need to back it up several times. You can check whether it is HD or SD in the lower right corner of Bitcoin-QT. Encrypting the wallet (maybe even changing the passphrase) will required a new backup!
and i also believe that bitcoin core (from 0.13 ?) is using BIP 32 deterministic Key Generation or in other words creates a seed (although i believe there is no seed words but just a master private key)
Bitcoin Core 0.13.0 and yes a master private key.
for you that you can back up and restore your private keys from.
Correct, although the tool for recovery has not been yet made available. It should be released soon.