If you live in a city, prefer anonymity or don't want bank hassles, the easiest option to acquire bitcoin is to make a face-to-face trade with a local seller.
LocalBitcoins is the primary site where such transactions are arranged and prices negotiated. The site also provides an escrow service as an added layer of protection for both parties.
There are security considerations for both buyers and sellers, especially if the trade is a sizeable one. Always meet in a busy public place, don't meet in private homes and take all the precautions you'd usually taken when walking around with large amounts of cash.
Remember, if you're meeting face-to-face somewhere, you'll need to have access to your bitcoin wallet. Whether it's a smartphone, tablet or laptop, you'll also need live Internet access to confirm the transaction.
If one-on-one trades aren't your thing, check out Meetup.com to see if your area has a bitcoin meetup group, where you can do it all in a group setting and learn a lot from the other members in the process.
Depending on the seller, you may pay a premium of around 5-10% over the exchange price for a face-to-face trade, for convenience and privacy. A reputable trader will negotiate the price before a meeting, but many won't want to wait too long in case bitcoin's value takes a dramatic shift.
Some sellers may let you use a PayPal account to pay, though most prefer non-reversible cash for the reasons described earlier.
It's also wise to check first if such trades are legal in your local area. There is also a slight danger you'll arouse police suspicion by exchanging cash in a public place, if they think you're trading something more illicit.