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.