Virtually any altcoin exchange would do since almost all of them support LTC. There are many exchanges out there including Cryptsy (which has already been mentioned) and Poloniex (which is currently my personal favorite). I've heard of Shapeshift.io and they seem to have good reviews although I've never used them myself. I understand that they work a bit differently compared to traditional exchanges.
Looks like I would need a Litecoin account?
Not sure what you mean by this but in order to actually own litecoins, you will need a Litecoin wallet. Just like the way you need a Bitcoin wallet if you want to own bitcoins. I wouldn't advise storing your funds in an online wallet that you don't have the private keys to and the only online wallet I'm aware of which grants you control over your private keys is Blockchain.info but they don't support Litecoin.
I would advise downloading the official Litecoin client
here or alternatively, using a thin client such as Electrum-LTC if you don't want to download the blockchain which can be found
here.
