I hate to mention this, but from my limited experiance with walking around underground, it seems to me that it doesn't work over the internet. It looks like it only works with people nearby, kind of like Pictochat.
I could be wrong. In fact, I hope that I am, but from looking through the instruction booklet, my small amount of Japanese confirms it. The WiFi is only for friend code battle/trade, battle tower, and the world trade thing.
If somebody does figure it out, I would love to hear about it.
And as for getting underground, you get this shovel set from the town that you fight either the second or third gym leader, right before bicycle road. They don't let you leave the town until you get it from the guy, so you will get it. To use it, find what looks like a big dip in the ground, and use it from your key items.