How do I spend my Bitcoins without waiting for Bitcoin-Qt to sync?
If you use an iPhone, give bread wallet a try, the Blockchain app is cool, too. On Android I'd probably use mycelium. 

All of those have nice and clean interfaces. Send some coins back and forth to get a feeling for it. 

It's better for the network to run a full node (=use a client that downloads the whole 20Gig blockchain) but you don't necessarily need to, to send and receive BTC. All the apps I mentioned before dont require you to download the blockchain.

Technically Blockchain (the company) is an online wallet, which means they hold and secure your private keys (= the keys that allow sending money) while with bread wallet and mycelium you hold the private keys.

Blockchain = the company

the blockchain = the public ledger of Bitcoin