Pokémon Shining Diamond and Shining Pearl have a lack of fire types during the game's start, so it might be a relief that you can find a Ponyta shortly after getting the second badge. Here's how to find it.
The lovable fire horse Ponyta can be found on Route 206 south of Eupolis and north of Ore City. It is under the bridge that takes cyclists through the area as there are no wild spots on the intersection itself. Simply go across the bridge, defeat all the trainers in your path and then, once off the path, head right to find some trees to cut down. Using Cut, cut down a tree to clear a path and enter tall grass.
You'll encounter some Geodudes and Kriketunes, but keep trying until you find the adorable Horse Pokémon. Once you are in battle with one, use relatively weaker attacks on it to bring it to a red bar. We used the Slash attack itself to get into that sweet spot as it's not a high damage move. Now use a sleep move to knock the Ponyta unconscious. Throwing a regular Pokéball worked for this encounter.
The original game also has Ponyta running around on routes 210, 211, 214, and 215, but we haven't tested it yet. We will update this post once we find out if this is the case.