How do I make my first purchase of cryptocurrency?
Download and install Bitcoin Core: https://bitcoin.org/en/bitcoin-core/
Create a wallet and receiving address there (it's better to use encryption when creating wallet).
Buy bitcoin on any available BTC exchange (Coinbase, Binance, etc.).


I'd like to keep my own wallet and not use a Coinbase type of service.

You can keep your wallet on your own, but to buy BTC, you will need to use service or find a person who wants to sell you ones.