How does one get started with the account claiming process? If there are multiple methods, which do you recommend for me?

Just go to your wallet in peakd.com they provide a frontend button that will create the operation for you. Click the down-arrow next to HivePower.

I know exactly what you mean about security and privacy, and I'm fully on board.
My system for buying and selling Hive accounts like NFTs will provide tons of extra anonymity to the network.
My other system for an air-gapped hardware wallet is also key.
I just need the funding and logistics to make it happen.