I know that you can't view both incoming and outgoing transactions of a cold wallet, but I'm still confused as to how I can see if my previous transactions I made have actually reached my cold storage address. I know I should use the view key, but don't know if I should use the view key for the cold wallet in simple wallet or in a block explorer.
The CryptoNote protocol allows you to verify that your funds arrived safely at your cold wallet address. This is done as follows. You can verify that your funds arrived safely at your cold wallet address using this tool or this explorer/tool. The former tool is created by luigi1111, who is one of the core-team members of Monero, and fetches data from Moneroblocks. The latter tool uses its own explorer to fetch data, namely Monero Explorer.
First, make sure to save the transaction hash shapeshift, an exchange such as Poloniex, MyMonero, or simplewallet provided you with. Fill in the cold wallet address, the viewkey that belongs to the cold wallet address (MoneroAddress provides this), and the transaction hash of the specific transaction. It should show you:
This address owns output x with pubkey: xxxxxxxxxxxxxxxxxxxxxxxxx for amount x.xx
The sum of those outputs should be equal to the amount that was sent. An example of how it should look like can be found here. Generally it is suggested using a test amount first to try out the tool and get a bit familiar with it. Furthermore, the tool will check if the viewkey matches the address. Thus, in case you made a typo when noting down the addressor viewkey it will show you an error message to prevent you from sending funds into limbo. Alternatively, if you made a typo when typing over the address or viewkey from either the printed page or the page were you noted down everything it will show you an error message too. Using this tool you can be certain that your funds arrived safely at your freshly generated cold (paper) wallet. Even if you are filling your "real" wallet, it is generally advised to first send a small transaction to test if "everything works". Furthermore, it is advised to first make a test wallet (e.g. 1 XMR) and restore that, so you can also verify with your own eyes that the funds arrived safely at the address.
Lastly, if you are looking for a guide on how to securely generate an offline cold paper wallet, it can be found here.
outgoing transactions of a cold wallet
For clarity, there are possibilities to do this. These have been discussed extensively before and implications have been posted in this thread. Furthermore, an important "feature" to do so has been merged into master a few weeks ago. From the pull request:
They are used to export a signed set of key images from a wallet with a private spend key, so an auditor with the matching view key may see which of those are spent, and which are not.
It should be noted though that key images only work in a adversarial way.