Unless you have some sort of contacts that you can sell to or buy from, your best bet is most likely finding someone willing to trade cash for bitcoins on LocalBitcoins. Depending on where you live that may or may not be easy, for example there is no one selling where I live and I would have to travel at least 200 km to buy or sell with cash.
Yeah it's all about your area unfortunately, if you live in an urban area you could go for
LocalBitcoins which is a pretty trusted site for exchanging with cash (and PayPal etc sometimes, but be careful with that because scams are always possible and it's P2P). If you don't, the first thing I would do is exchange with someone on the forum for PayPal (they might know your name but it's very unlikely they'll use the information), but if you use PayPal you need to choose a really good person because PayPal transactions are reversible and everyone will think you're trying to scam them.
A few credit card sites don't need your name, namely CoinMama for small amounts (under $150) and if you live in an urban area you could use a Bitcoin ATM, but them having your credit card details could make me pretty insecure as well.