The only thing you can do to stay anonymous is using decentralized exchanges so you will avoid to send your ID to third party. My suggestion is you search for your local seller who sell Bitcoin and receive fiat from your bank.
Bank? That is not much better as the bank is also a third party. They may already have your ID but that is not important in this topic. You should avoid using your bank for these activities if you can.
Frankly speaking there are even some exchanges that doesn't require ID to buy BTC using credit card.
Which ones are those?