How do I spend my Bitcoins without waiting for Bitcoin-Qt to sync? I purchased $50 from ExpressCoins and now it's transferred to the address, but I hadn't realized that to use your program, I have to download 20 gigs of information.  Is there a way to access my coins without doing that and transfer them to the market in order to spend them?  I can't use the address at another wallet that doesn't do that?

A clear step-by-step would be a godsend at this point, and looking for threads and faqs to read has just confused me with more vague answers and piling on new programs and terms.
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