What trust am I placing where when I use Electrum? I have read that Electrum never is given your private keys (which makes sense), but I'm still unsure what sacrifices it makes in order to not require you to not store the entire ~14GB blockchain.

So is the gist of it "you make transactions on your own using your information, but we watch the blockchain and tell you anything relevant that happens"?  Am I understanding this at all correctly?

If so, who is "we" in that?  And can't whoever I am trusting lie to me about how many bitcoins I have?
"We" are volunteers running Electrum servers. There's almost no trust in them, the absolute worst they can do is feed you false negatives about the balance of your wallet, or they can also choose not to broadcast your transactions and pretend that they have. You trade some privacy in that they can see the contents of your entire wallet and associate it with your IP address. The client supports Tor connections if this is an issue for you. 

The blockchain is 22Gb now, actually.