If you're looking for a super-easy way to get a credit card, check out PayPal. They have something called the "PayPal Debit Card", which allows you to fund a debit card directly from your PayPal account.
The debit card can be used at most ATMs to draw cash, or can be used at just about any place a MasterCard can be used. (Which is almost everywhere.)
First you have to get a PayPal account, which is free to set up.
Go to the New User Registration, then click the Get Started button under "Personal".
Enter all the requested info, and then continue to follow the instructions given to setup your account. I'm sure you'll have to receive an e-mail and click a link in it (or something like that).
After establishing your account at PayPal, you need to establish a means of getting money from you into your PayPal account. If you don't have a credit card, then you do that with a bank account. Log in to your PayPal account, click the link to edit your Profile, and then there is a section for managing the financials. Click the link for adding a new bank account.
You'll also want to become familiar with the ways you get more spending power with PayPal. To do so, you need to verify your account, and their site has all the instructions for doing that.
After everything is all setup and working, you can apply for their debit card. The link can be found on their site.
Once you get the debit card you can use it most places like a credit card. For example, Lottery Post's credit card system accepts Visa and MasterCard debit cards just fine, so you would be able to use it here. The gift shop would also be able to accept them.
Just be very careful to keep track of what you're spending, and do not attempt to spend more than is in your account, especially if you're trying to establish credit.
Also, allow for some time to get everything set up correctly. Because you are trying to get started without a credit card, you can't expect to get everything setup and working overnight. It may take a week or two.
Good luck!