Not sure if you're still looking, but my first "slimmer" wallet after using a trifold my whole life was the dash wallet off Amazon. It's not super nice leather or anything, but it's slim, will fit ID, insurance card, a couple credit cards, and some cash. It'll last a year or two.
I've since gone even further down the minimalist hole with a Saddleback single pocket leather wallet, but the dash one is what I'll recommend to start off with if you wanna start slimming down the wallet.
Dash 3.0 is solid. Limited room for cash tho. http://www.amazon.com/dp/B00YD8VRFM/?m=A23JPW825T6PO5&ref_=pe_823600_114105210