I'm guessing "the guy" was at an AT&T store. Bottom line: this is wrong.
You cannot transfer contacts on your SIM card because you cannot use your old SIM card in an iPhone. Your iPhone will accept contacts from your address book on your PC/Mac, or from certain online services like Yahoo or Gmail.
If you have a way of managing your contacts on your old phone on your computer (most phones give you a proprietary app for this), you might be able to transfer them to a Mac, PC or online address book, and thence onto your iPhone. If not, you can at least copy and paste them individually to a compatible address book. Because you keep your old SIM in your Razor, you will still have all the contacts on there.