Can I exchange BTC from CoinBase?

I have read the articles in the BitCoin website and I thought of implementing the BitCoin payment method in my website.

I registered at a payment processor website called "CoinBase". Unfortunately my country is not supported for the Buy & Sell feature.
I am thinking of accepting payments and later using some sort of exchange website to cash out. If I do start accepting BTC should I store them in the CoinBase wallet or use something like BitCoin Core? I am asking this because I heard that some BitCoin services steal the BitCoins from their customers and run off. Other than that, storing them online is probably more comfortable.
Could I get some suggestions on good exchange websites?

Thanks in advanced.