I'm kind of on the same boat as you, so here's what I did:
Get a store credit card (Macy's etc. wherever you shop..) Gas card or any other VERY BASIC card and use it. WARNING: do not charge more than you can afford, otherwise you will fall into debt and pay very high interest!!! This is supposed to be common sense, however, a lot of people ignore this very important note.
Moving on... then, make your payments on time, etc. but do not spend more than 30-35% of your available credit. For example, if your card limit is $1000, don't charge more than 300, its the same as if a stranger asked you for money and you told him/her the most you can give is $100, would you feel comfortable giving $10, or $95... same concept. Money can be risky
After you build credit with your "Training wheels card" then apply for an Amex Green, since you can't upgrade from the Zync, though if the Zync is more of your cup of tea, then stick with it.
Wallet: Amex BCE & Chase Freedom. (Credit Card dieting since 2014)