Before you go trying for a Discover card, do you have ANY credit? Your first step should be getting copies of all of your credit reports and seeing what's on them. As a NY resident, you're entitled to a free copy of each annually.
Once you have copies in hand, you're ready to start building. Start small and don't overextend. The best credit cards to start with are low limit store credit cards. They'll let you build credit without spending more than you can afford. Get one or two max, and pay them off right away. Do not miss a payment.
Every six months or so, call the companies that you have cards with and ask them if they'll increase your limit. This will enable you to have a higher credit line, and other companies will see you as someone who can manage a bit more credit. (To a point, but you don't need to worry about having too much credit right now).
Discover is actually tougher than some cards, and you will probably be best off looking at a Capital One credit card as your next step. It's a long road, but if you don't go and screw up, you'll eventually get your Discover card. (Although by the time you can get one, you'll probably want an Amex...)