What happens if I don't "pick up" bitcoins with a Bitcoin client? Say I want to save ten bitcoins could I send them to myself and just not 'pick them up' with my client? In a way would I be storing the bitcoins in the blockchain and when I want them just start up the client? Or would it be possible for someone else to 'pick them up'? Or are the private keys already in the wallet file as soon as the payment is 'sent'?
Everything is already stored in the blockchain, and there is nothing to "pick up".

The blockchain stores: "A generate a block (25 btc now)" and "A sends 2 btc to B".

When you want to send bitcoins, you proof that you are "B" and sign a message saying "I own these coins (A->B) and want to send all of them (2 btc) to C".

Now "C" owns 2 bitcoins.

(This of ofc simplified alot, e.g. I am missing confirms/etc but you really need to understand the basics before you start learning the advanced)