1. Do you simply go to the main Electrum website and download a new wallet?
Yes, go to the
Electrum downloads page which will give you the download to the latest stable Electrum package. You should be able to install the latest binary and have it update automatically (at least this is the case on Windows, and perhaps Linux if memory serves correctly).
2. If so, does the new wallet transfer the btc over automatically?
Your wallet file is stored on your hard drive (in the appdata folder by default on Windows). So long as you show Electrum where your wallet file is (or have left the location for it at it's default), Electrum should still accept the wallet and use it.