Nintendogs Walking

Walking
All dogs love a good walk, and you should do your best to get your dog out of the house at least once a day. You can take them out as often as you like, but you'll need to wait at least 30 minutes between walks to ensure that your dog doesn't get tired out. Walking your dog will help it build up stamina, which will allow it to go for longer walks in the future, and will also let you find presents, meet other dogs, or visit the park or other areas of the town that you dwell in.
To begin a walk, select Go Out - Walk. You'll need to draw a path with the touch screen through the neighborhood, but you initially won't be able to go too far; your dog will get tired quickly, and thus you'll have to return to home before it gets worn out. You'll notice a small meter in the lower left corner of your touchscreen while you draw your path; this is your dog's stamina meter, and you won't be able to exceed it when drawing your path. It's a good idea to try and make your walking path as long as it can be when taking your dog out; if you do so, you'll increase its stamina and allow it to travel much further over time.

If this were real life, you'd be calling the bomb squad right about now, but in Nintendogs, someone has helpfully left gift-wrapped items all over town!
There are a few events that can occur while walking, most of which are represented on your map with a question mark icon. Try to go through as many of these as possible, as you may find presents that you can sell for other cash, or meet new dogs that your own dog can become friends with!
Elimination
Your Nintendog has to go to the bathroom somewhere. Thankfully for you, it's perfectly housetrained when you get it, and will wait until you go out for a walk to poop. Every so often, usually once a walk, it'll rear up and drop its business on the sidewalk. If you want, you can click on the poop to magically transport it into a baggie, or you can just leave it lie and hope it gets in Mr. Krunkle's shoes when he goes out for his morning jog. He's mean.
Your dog may also occasionally pee. You can't clean this up, but let's be honest; would you really want to, even if you could? Every dog's got to mark his territory now and again, and indeed, the spots where your dog pees will be marked on your map with a little blue spot. If your dog hits one of these spots on a future walk, it'll usually refresh the marking by peeing again, especially if another dog tried to overwrite it with its own urine. This doesn't have much of an effect on anything, so far as we can tell, but your dog might be happier if it gets the opportunity to mark its territory in a wide variety of places.
Trash
Every so often, you'll come across trashbags on the sidewalk when you're walking. If your dog gets a whiff of some of that sweet, sweet garbage, then it'll probably try to eat some of it! Be sure to yank the leash forward to the right edge of the screen to prevent this from occuring. It's not necessarily a bad thing if your dog gets a banana peel or two down its throat, but it's not something you really want to encourage.
Meeting Other Dogs

Meeting a friendly dog will make your own dog feel happy. Awwwwww…
When your dog meets up with another dog, a few things can happen. In most cases, the two dogs will start out by barking at you while you desperately hit on the other dog's owner. There's precious little you can really do to influence your dog's behavior here, but you can move the leash around to goad your dog into approaching the other dog if you wish. The two dogs may eventually settle down and get friendly with each other by nuzzling, or just keep barking until it's time to go, or actually start fighting by lunging and attempting to flip each other over.
In general, it's worth waiting around when you meet another dog to see what happens. If the dogs get friendly, your dog will feel better about himself; if they remain curious about each other (indicated by nonstop barking), then no harm done. If they start to fight, though, you'll probably want to break them apart by clicking on the back arrow and leaving the dog in the dust.
Landmarks
In addition to just wandering aimlessly, you can also visit new buildings while you're out and about on your walk. There are a couple of parks near your house, as well as discount shops and a gym which can be used to train your dog for agility contests.
The parks here are going to be useful for training your dog for the disc competition, or just to get it out for some company with other dogs. There are two parks in the neighborhood, one empty and one with a couple of dogs playing in it. Note that the dogs move from park to park, so if one's empty one day, it might be full the next! Disc training is best at the empty park, but playing around with the other dogs at the full park can be amusing as well, especially with items like the pull rope which more than one dog can use at the same time.
The gym (there's only one of them) will be represented by a building with an icon of a dog jumping over a hurdle on it. If you visit the gym, then you'll be able to put your dog through the ringer and mercilessly belittle it until it finally learns how to jump over a hurdle. There are also hoops coated with flame, but you won't be able to access those until you're ready to sell your dog to the circus.
The discount shops are a bit too distant to walk your dog to when you first get one, but over time, you'll be able to reach them, and you'll be glad when you can. Although the main shopping area that's available from the "Go Out" menu offers everything you absolutely need to take care of your dog, the discount shops will have many of the same items at much cheaper prices, as well as odder items like keyboards, new accessories, or other surprising items. The inventory of the discount shops changes over time, so be sure to visit them whenever possible to find new stuff!