So if I use the Armory client offline to generate a new Bitcoin address, I can immediately start sending bitcoins to that address right? Without it ever connecting to the network? How does this work, I don't understand where the bitcoins are stored.
Would I then be able to add this address to a Blockchain wallet and check the balance?
A paper wallet is just a printout of the private key associated with a given address, correct? What does this have to do with the wallet then?
If I create a watch-only address on Blockchain, how do I find the private key for that address so that I can store it somewhere else? Is this good security for a savings type of account then?
I am trying the find the best way to make a secure savings account, but the tutorials I am reading are not explaining the fundementals of how these parts all work together.
Any help would be appreciated.
Bitcoins are "stored" on the network; that is the only place they exist.
To spend them (i.e. transfer them) you need a private key. There is a mathematical relationship between the private key and the bitcoin address. You can safely make an address public; it is deposit-only; no-one but you can spend as only you have the private key.
A wallet is just a fnacy name for a collection of private keys. Private keys are what are stored on your computer, or on a piece of paper.
You can input a private key into most bitcoin software, including e.g. blockchain.info, and they will then show you the balance that that enables you to spend, and permit you to transfer the coins.