It's my understanding, when using bitcoin, it's advisable to generate a new address for every transaction. Is this also the case for Ethereum, and if so, how do I do this with Mist?
Bitcoin is UTXO based, so you can use an HD wallet(BIP32) to get a new address for each transaction to protect your privacy. However, Ethereum is account based and each account has a single address, but it is possible to implement an HD wallet for Ethereum. I think there is already such wallets as Jaxx which provide such ability.
So to answer you, if your privacy is crucial to you, please use an HD wallet for Ethereum. Mist don't offer this option.
Read more about HD wallets https://github.com/ethereum/EIPs/issues/84