Once you're in a Nintendo zone just go in game and head to the post office. Talk to whichever pelican sister is in charge and ask about a present. There should be like a saving sequence (I think) and after it's through she'll hand you a present. I have heard a lot of complaints about Nintendone Zones not working though. Hopefully yours does!
I tried recently for the Red Oni Mask at a McDonalds, and it didn't recognize the zone.
It's spotty at best. A really underwhelming design choice by Nintendo. Just like the Spotpass feature, it only makes more sense in Japan, compared to mostly anywhere else where it falls flat.
I went to my local Best Buy and it worked no problem and I'm not too tech savy and had no instructions. Just went to get present from post office (took a chance) and it took 2 secs.
I forgot to say you dont have to leave home for a Nintendo Zone item. You can change your Wi-Fi's name to a name used by a Nintendo Zone to get it from home. You can also do this with your cell phone if you have a hotspot feature. Change your phone's name and use as hot spot. The thing about places like McDonalds and Best Buy is that they sometimes change their Wi-Fi name so they stopped being recognized by the 3DS as Nintendo Zones.