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 &amp; 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.
It's generally a good idea to hold bitcoin in a wallet you control vs exchanges like Coinbase. I wouldn't use Bitcoin Core however as that is primarily a consensus server at this point. Quite a pain to run. Check out other standalone wallets like Airbitz (disclaimer: co-founder) or Mycelium, Electrum, etc.