I have an original DS and my daughter is interested in some of the new games like NintenDogs. I've looked everywhere, but I can't find any information on this. Will the new DS games work on my old one? I understand that games specifically for DS3 and DSlite won't work, obviously.
I just don't want to spend the money on new games if they won't work.
Yep, they will work!
All Nintendo DS games will work on any Nintendo DS system, whether it is the original DS, the DS Lite, or a new system such as the DSi, DSi-XL, or 3DS.
As you noted, 3DS games will of course not work on a DS or DSi of any type.
You may think that is obvious that DS games can be played on the DS, but noooooooooooo, cause there are Wii games that can't be played on the Wii (like DDR).