This step by step guide will walk through the exact method you can take to buy LTC.
Firstly you will need accounts with the following websites:
Step 1. Buy Litecoin
Once you have an account with Coinbase you can purchase some Litecoin, the screenshot below shows where you can do this by clicking the Buy/Sell button and then selecting Litecoin:
Congratulations you should now have LTC in your Coinbase wallet.