They won't know who you want to send the btc but they can track the transaction you made on coinbase since they collect data. Why did you use coinbase in the first place?. There are much better wallets out there like electrum.
.................go for Electrum wallet or Trust wallet.
Electrum is surely a good choice, but I wouldn't recommend trustwallet. Trustwallet gives you access to your private keys. But since it's close-source, there is no guarantee that you are the only one who has access to the keys.
Electrum is certainly a good choice when you want to use a noncustodial wallets. Trustwallet is fine in my opinion base on the time I am using it and also my experience using it plus it is also a noncustodial wallet but one thing that many people doesn't like because Trustwallet is a close source that will make the one who use Trustwallet think who might have the private key or mnemonic phrase of the wallet. It's already more than a year since I am using Trustwallet but the way I use it is to receive and send right away instead of receiving and storing btc. So far, I didn't notice anything that someone have access to the wallet or a transaction happen without my knowledge.