Why is the private key generated first in public key crypto? I'm still new to public key crypto. I'm trying to understand how it is that the private key is generated first in public key crypto? I would have expected the private and public key to be generated at the same time, as they would depend on each other.

For example, in a Bitcoin wallet, I can choose a private key (brain wallet), which will then be used to derive my public key (then my BTC address).
In general, the public and private keys are computed together.

For some schemes, the public key is computed from the private key. ElGamal is an example. (The system parameters include a suitable cyclic group $G$ with a generator $g$. Choose a random exponent $a$. Compute $y=g^a$. The public key is $y$, the private key is $a$.)

For other schemes, this is not the case. Some variants of RSA are examples. (Choose primes $p$ and $q$. Compute $n=pq$. Choose (somehow) $e$ relatively prime to $(p-1)(q-1)$. Compute an inverse $d$ of $e$ modulo $(p-1)(q-1)$. The public key is $(n,e)$, the private key is $(n,d)$ (or something equivalent). Notice (a) how the public key is completed before the private key, and (b) it is not in general possible to quickly compute the public key $(n,e)$ from the private key $(n,d)$.)

(For completeness: Note that of course, the private key cannot be quickly computed from the public key. If it could, any adversary could.)