I used Shapeshift to convert BTC to XMR, and then had the XMR send to my Monero Desktop Wallet. This was the first time I used this. I noticed I had to sync, before the balance will show, but I don't want to download the 30GB blockchain because I'm on capped data. This was the default and I used the receive address from the wallet to give to Shapeshift. Now, I read about and maybe I can avoid downloading the 30GB if I use a moneroworld node? But once I set up that remote node, the receive address is different, so do I have to just download the 30gb blockchain data before I can receive my funds?
If you had used the Monero GUI (desktop app) to create your Monero receiving address, your address will not change if you decide to instruct the GUI to use a remote node.
Syncing with a remote node means you don't have to download and verify the entire blockchain, but you still need to download part of it and it still requires your PC to do some very heavy computation which could take several hours. The use of remote nodes will not make anything 'instant', in the way that using a Bitcoin light wallet would. This is because of Monero's "stealth address" privacy mechanism that requires you to scan the blockchain for incoming transactions. Your wallet will not reveal your Monero address to the remote node.