The most used way to obtain some bitcoins is to simply buy them, there are many sites that allow you to buy and sell bitcoins, from currency exchange, with or without bank account, using the most current systems of money transfer like SEPA transfer, Credit Card, PayPal.
But please, be careful with your money. When sending money to an exchange or seller you are trusting that the operator will not abscond with your funds and that the operator maintains secure systems that protect you against theft.
It is recommended that you obtain the real-world identity of the operator and ensure that sufficient recourse is available.
Because Bitcoin services are not highly regulated a service can continue operating even when it is widely believed that it is insecure or dishonest and webpages recommending them.
Exchanging or storing significant amounts of funds with third-parties is not recommended.
Market exchanges are those in which buy orders are matched with sell orders, so the price offered depends on the market. If a service quotes a price then it is not operating a market but instead is a broker that buys and/or sells coins.
Below is a list of some of the most popular sites where you can sell or buy bitcoins.
BleuTrade A new large multicurrency exchange and trading platform. Sign up for an account, verify yourself and you can purchase a large number of cryptocurrencies.
Cex.io Another large multicurrency exchange and trading with the option to purchase also ghs for cloud mining.
BitStamp Buy and sell bitcoins with other users through an online marketplace.
CoinMama Fast and easy way to purchase bitcoins with Western Union, MoneyGram and credit cards.
BTC-E A platform for individuals interested in buying and selling Bitcoins using an assortment of world currencies.
But the best way to anonymously buy or sell bitcoins is to find a local buyer or seller so you can trade directly with another person, for this purpose I suggest:
People from different countries can exchange their local currency to bitcoins. The site users post advertisements where they state exchange rate and payment methods for buying or selling bitcoins.
You reply to these advertisements and agree to meet the person to buy bitcoins with cash, or trade directly with online banking.
Bitcoins are placed in LocalBitcoins.com web wallet from where you can pay your bitcoin purchases directly.