First, you need to register an account. If you haven't already done so, click here.
Head over to the main buyer search page and select how you want to pay.
You will see a table of all the applicable sellers, like the one below. Find one with your desired payment method and cost, and hit the buy button.
If you are buying with cash, make sure the seller is near you. You or they will have to travel for a meeting.
You will see all of your trade information, as well as a chat window to talk to the seller. Say hi, and organize how you will pay. At this point, they will send Litecoin to the escrow account. It takes 8 blocks, or about 20 minutes, to confirm, so be patient. Once the transaction is completely funded, send the money and click the mark payment complete button. If you have any questions at any time, ask the seller! They should be very knowledgeable and friendly.
Don't send money or mark the payment as complete until the escrow transaction has been funded. The buy process will tell you when you can safely make the payment.
The Litecoin should be added to your account now. If everything went well, give the seller a positive rating. This helps them get more customers. If there was an issue, click the open a dispute button. From here, you can explain to the LitecoinLocal staff what happened, and they should get on it soon.