What I'm struggling to understand right now is after keys have been generated & money has been transferred how would that be registered on the blockchain in 2010 without the use of the Bitcoin Client.
Coins "live" and move only on the blockchain, never anywhere else. The purpose of any wallet software or hardware is to hold and manage public (watch-only) and/or private keys, with latter to sign transactions, so that you're able to move coins for which you've appropriate private keys. Additionally a wallet shows you the balance of all coins your private/public keys control. Very loosely summarized...
If you want to entertain yourself with learning some Bitcoin knowledge, feel free to hop over to
https://learnmeabitcoin.com, great site to scratch the basics or deep dive to serious technical stuff and more!