How to send bitcoins from an online wallet to a blockchain.info wallet? I bought a small amount of bitcoin. They are in an online wallet. I would like to send them to a more secure wallet, my blockchain.info wallet. I have a few questions about the process.

From the online sending wallet. I must enter my blockchain.info receiving address. Enter the bitcoin amount. Then hit send. Am I missing any steps? Do I need to do anything with private keys or will that be done automatically? 

From the receiving blockchain.info wallet. After the bitcoin balance shows up in my wallet, am I required to do anything? Do I need to do anything with private keys? Should I back up the wallet afterwards?
That's all there is to it.  The wallets handle the private keys.  

It you have access to the private keys for the online wallets, it is always a good idea to export them and save them in offline storage (not on your computer).  That way, if the online wallet goes away, you can still recover your coins by creating a new wallet and importing the private keys.

If you don't have access to the private keys, a better option would be to create your own wallet (either on your computer or offline on paper).  Send the bitcoins to this wallet instead of an online wallet.  This way you are protected if the online wallet disappears or is hacked.  MultiBit, Electrum and Armory are all good wallets, so pick one that meets your needs.  You can find more information here.