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?