There are a two main ways to do this.
1. Use Bitcoin ATMs (limited amount you can transfer from fiat to BTC and also has terrible exchange rate)
2. Localbitcoin (Transfer your cash for bitcoin with another person, which can be risky so be careful)
Both options are not 100% anonymous. Most of the ATMs in my place are having cameras installed. The chance that you may be identified from the camera footage is quite remote, but the risk is still there.
And regarding LBC, most of the buyers will only deposit the amount to your bank account through online transactions. It can be traced. Even if you deal with them in physical cash, setting up an LBC account requires ID verification (or at least a phone number).