I just went with a friend to the Asian Supermarket today and bought a few snacks and candy. Mmm. Yummy. So if you're ever in an Asian Supermarket, I highly recommend the following candy. You can buy these candies online too, but they're usually more expensive online versus getting them at a store.
Kasugai Gummy Candy

These fruit flavored gummies are delicious. The flavors are very true to the fruit, I believe the Kiwi ones actually have kiwi seeds in them. My favorite is the Lychee (Litchi) gummies. I grew up eating Lychees and I can honestly say that the flavor is extremely close. Lychees are a very sweet, flavorful, and juicy fruit. It's one of my favorite flavors. Unfortunately there aren't very many American Lychee candies.
The gummy itself is round and packaged individually. They are not sticky and will not glue themselves to your teeth. They are meant to be enjoyed a few at a time, but it's very hard not to eat the whole package in one sitting. I try hard not to.
Flavors: Lychee (Litchi), Strawberry, Grape, Kiwi, Orange, Melon, Peach, Pineapple, Apple, Muscat
Price : Around $2.30 in stores. You can buy them
here for a reasonable price.
Frutips

Frutips is a candy made by the Nestle company. They are fruit flavored pastilles (a harder type of gummy). They come in Black Current, Strawberry, Lychee (yay!), and Mixed and are made out of fruit juice. They packaged in a tube or bag.
The texture can be describe as a mix between a sour gummy candy (although it's not sour, but it has a coating of crystals) and a gumdrop. It's not as soft as a regular gummy and it might stick to your teeth every once in a while. The flavor is nice, I like the Strawberry ones. Yum.
They cost around $2.50 or so in stores. I heard they are also popular in some parts of Europe too.
Ramune

Okay, so this isn't really candy, but it's still fun to drink. Ramune is a Japanese soda with a unique way of opening. In order to open the bottle you must force a glass marble stopper out of the cap and into the neck of the bottle. The neck has a small area to catch the marble, there is also a stopper on the side of the neck to prevent the marble from rolling back into the cap when you drink (however it's only on one side of the neck and occasionally the marble will slip past).
The method of opening the bottle may be a little pointless, but it sure is fun (and very very messy). The soda comes in various flavors including Original ( a sprite-like flavor), Strawberry, Peach, Orange, Melon, Raspberry, Lychee, and a few others (including a Hello Kitty themed bottle).
I've heard stories about people breaking the bottle after finishing the soda and keeping the marbles. Also when you open the bottle, you can't close it again, so beware of spills.
A bottle usually costs around $1.30. You can buy some online
here.
Calbee Shrimp Flavored Chips

Don't knock it until you've tried it. I personally don't think they taste like shrimp at all, but they are yummy. And they're not really "chips" per se, but they're more like Cheetos. They are long in shape with linear grooves down the side. They have a nice salty flavor (not briny at all) . It's kinda hard to describe. But they are yummy nonetheless. They also come in a variety of flavors like Wasabi (which I'm none to fond of).
I've been eating these since I was a little kid. The packaging has pretty much remained unchanged.
If you decide to give these a try, I recommend getting a small snack sized bag. That way if you dislike them you won't be wasting a lot of food. The snack sized bags are reasonably priced too, around $0.65 or so.
Conclusion
If you don't have any Asian Grocery stores nearby and do not want to order online, I would recommend stores that carry Anime merchandise (like Suncoast video/entertainment stores). They usually carry a few snacks like these (mainly Ramune).
Stay tuned for Part 2 and more snacks from the far east.