You could buy bitcoins from any reputed exchange.But don't leave your bitcoins there.Download a new wallet ,for example from Electrum.org website and install it in your desk top.Don't ever give your private keys to any one giving offers that your investment will be doubled.
You could buy bitcoins from poloniex or bittrex exchange.Never leave your bitcoins in any exchange as it may get hacked as it previously occurred in Mt.Gox exchange.
I'm usually use local exchanger for convert my btc into fiat and for secure wallet, I recommend electrum also ( offline wallet )
I've been using over 6 year and I was use blockchain over 3 years until they disable pay to many feature.
So yeah.. it's recommend to save at offline wallet rather than online / exchangers