It seems that you are a very skeptical kind of a person because all you do before are to watched the scene of Bitcoin instead of trying to start investing with it I can't blame you though as many people are like you and I once like that when I am starting my early stage with it.

Blockchain.info and Coinbase is an exchange for Bitcoin that can considered also an online wallet so you will not be holding your private keys there and you are entrusting your Bitcoins to them if you start investing, if you want you can install a desktop/software wallet to your computer like Bitcoin Core, Electrum, etc. as you will hold your very own private keys, I am using Electrum wallet since I started using Bitcoin though and I recommend it also because it is convenient to use for me.