Much ways you can choose if you want buy some cryptocurrency safely.. I think safest way #1 is buy via CoinBase , if your country are not supported on CoinBase to buy with Credit Card there.. maybe you can try via exchange , most recommended is Binance.
Once again, if your country not supported on Binance exchange to buy via Credit Card.. try BitStamp.
The op want buy cryptocurrencies anonymous because he won't the identity show up to the exchangers, so the market of cryptocurrencies does not suitable for him. And i think he can buy cryptocurrencies on decentralize application exchange or he can buy cryptocurrencies on local place, they will protect his identity from the public. I think it is good solutions for buying cryptocurrencies to him than traditional cryptocurrencies market.