When are you charged a withdraw fee? I'm a bitcoin newbie and think that I am missing something.  I'm based in Australia although I'm not sure if that matters.  I would like to buy and hold for an extended period some bitcoin and ether.  I would prefer not to use the exchange's wallet or storage.  Will I be charged a fee to withdrawl and use my own cold storage?
There isn't really a notion of "withdrawing" from cold storage.  Bitcoin isn't a bank and you don't make deposits and withdrawals.  You have some coins in an address, that's all, and you can spend them any time you like.  The only difference with cold storage is that you are storing your private key on a piece of paper / USB stick / offline computer / etc, so you will have to do a little more work to spend the coins.

Spending coins from cold storage just requires a transaction, and every transaction requires a transaction fee.  But this would be the same fee no matter whether you were keeping your keys in cold storage or in a web wallet or on your smartphone.

(You might find older resources suggesting that coins which have stayed in the same address for a long time can be spent without fee, due to the transactions being higher "priority".  This is not true anymore due to the current shortage of block space.)