So there are different levels to Bitcoin ownership. Hopefully explaining like this will help.
1. Buy Bitcoin/crypto from an exchange, leave the crypto on the exchange.
You own this crypto in the same way you own your money in a bank. That is to say that if the exchange folds or if anything goes sideways, they have legal control of the crypto and you lose it. Outside of this counter-party risk this is a fine short term solution. I would suggest doing this if you think that you'll be willing to learn more about how this all works once you have some skin in the game
2. Buy Bitcoin/crypto from an exchange, move it to cold storage.
You want to buy Bitcoin, but don't plan to do anything it but HODL. Get yourself a hardware wallet (Trezor or Ledger are the oldest and most trusted). They'll come with directions to get it set up. You're essentially going to get an address from this device and send the Bitcoin you buy from the exchange to this address. Your device is the private key for this address and as such nobody can touch your coins without this device.
3. Buy Bitcoin/crypto from an exchange, actively use it for trading/DeFi
If you're asking this question don't do this, but FYI, there is something called a hot wallet (ex. Metamask) that holds Ethereum (ERC20) tokens you can use a number of different methods (Wrapping BTC, or just trading with ETH or other alts) to engage with the burgeoning decentralized finance market. You can earn interest, stake certain coins for income, get collateralized loans, become a liquidity provider for a market, etc. This is all very new and quite risky at this time, but its something you CAN do with your crypto.
There are a number of solutions in various stages of development that will make buying and holding crypto easier and safer, but for now IMO you should either keep a small amount of crypto on an exchange to learn with, or get a hardware wallet and just store it all there.
As for the exchange process you'll go through the KYC process, ACH/wire them money, once it clears you can set a market/limit order for your Bitcoin. Once purchased you can choose to send the coins to an address through the exchanges interface. Its not too complicated once you get on an exchange.
You can also just buy BTC via CashApp if you want, although I'm not certain that they allow you to transfer to a different wallet.