I personally use Blockchain.info so I recommend it. It's easy to use.
And like @TMAN said above, I don't recommend using an exchange website as your wallet.
It's not like the Blockchain.info wallet is any different in functionality than an adress on coinbase/poloniex really...
There's no direct access to your private keys, ...
That is not true. Blockchain.info is web-based, but it gives you, and only you, access to the private keys.
Correct, you have the private keys with Blockchain.info.
If you want to trade regularly, I'd advice keeping some coins with your chosen exchange, for example coinbase, as there is cost involved transferring between wallets.