It is highly recommended that you not use a web wallet, especially if you have a large amount of Bitcoin.
I suggest that you use Electrum on your computer. You can create an Electrum wallet and send the Bitcoin from blockchain.info to Electrum.
To backup your wallet, all you have to do is go to File > Save Copy and save a copy of the wallet file to a safe location. You will also receive several (13 I think) wallet seed words. Remember or write down these words some place secure as you can use these words to restore your wallet.
You are right, but I still believe in private wallet for cold storage if you have many bitcoins on it. Of course if you use private wallet , you must spend many days to download and install blockchain but this fact is safe and necessary in the future