So bitcoin-qt takes part in the BTC network in some way when it's online? I thought it was just a wallet. Does electrum do this too? Are there any BTC apps that are just a wallet?
bitcoin-qt is a wallet. it also runs a full node, meaning it hosts an entire copy of the blockchain and relays transactions. this supports the network and all that.
electrum is a lightweight client meaning you do not have to download the entire blockchain.
if you just want to have a wallet go to bitaddress.org and print one out, send your BTC there and don't worry about your bandwidth usage.