What happens if I don't "pick up" bitcoins with a Bitcoin client?
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)