About to open my first wallet
Some online wallet services might require your id, however, if you create a wallet on your local computer there is no personal information required at all.

The balance of each address is public knowledge at all times, but neither is the owner of an address,  nor the fact that different addresses belong to the same wallet.

Some of this information can be gained by analysis of payment patterns or after information about ownership was revealed publicly. For example, addresses can be deduced to probably belonging to the same wallet, if both of them are used in the same transaction as inputs.

If you use big exchanges with KYC policies to trade your bitcoins for Fiat,  they will be able to tell on you. You might be able to make local exchanges for privacy.

In conclusion,  with sufficient know-how and micromanagement you can probably keep your total balance secret.