I always seem to come out of the wordwork for video game questions.
Anyway.
I'm not shopping right now so I can't tell you deals. But as far as the different types:
DS Lite: This is the oldest version out there and will likely be the cheapest. It's called the Lite in that it's lighter and smaller than the very first DS they came out with about 5 or 6 years ago. It plays all normal DS games, and it has a slot in the bottom to play older Gameboy Advance games. Given your daughter isn't an older gamer and won't have a larger game library to worry about being able to play, that second slot probably isn't a big deal. It can play online with some games that support it when you configure it to your home router.
DSi: This is newer than the DS Lite. It no longer has the secondary Gameboy Advance slot, but it still plays all normal DS games. The two major new features here are a camera to play with (I believe there are little games to mess with your pictures, that kind of thing) and the ability to buy games over the internet from the DSi store. There are a lot of smaller developers that put out their own little games, that's the kind of stuff you'd find in the store. But most kids just like the camera.
DSi XL: This is the exact same thing as the DSi, except it's about 50% bigger. Your mileage may vary if you think it's good or not. I think it was made for the older gamers so they could see, and to me it's far too clunky to be all that portable and I think the graphics of the games look stretched out and bad. Some people with younger children report the larger size is easier to manage, some say their kids think it's too big. If you're interested in this one I would recommend going to a game store and having a look at it in person. It's really quite big.
3DS: This is the newest DS. As the name suggests, it plays 3D games without the use of glasses. It's really quite cool, though it's not in my budget to upgrade from my old DS so I don't have one yet. It has the same camera functions as the DSi except it has two cameras so it can take 3D pictures. It also has the online store. As far as the 3D goes, there have been a variety of opinions back and forth on whether or not its good for childrens' eyes, or if it causes headaches, etc. You should know it has a switch on the side that controls how strong the 3D is, and the 3D can be turned off entirely making it like the normal DS with the 3D option if you want it. The 3DS can play both normal DS games and the new 3D DS games.
As far as price point goes, the DS Lite will be cheapest but you'll likely need to get a used one. I don't think they make them new anymore. The DSi will be next up, and I've found that the DSi XL and the 3DS are about the same price.
The other question about the dog game. You are thinking of Nintendogs, or Nintendogs + Cats in 3D. Both let you take care of a virtual dog (or cat) who you can pet with the stylus, teach to do tricks by voice command, etc. Both are lots of fun, but make sure you get the one that goes with your DS. Don't get the 3DS one if you are buying a normal DS and if you're getting a 3DS, get the 3D one because the effects are quite good.
Hope that helped some! Let me know if you have other questions.
