Other than go fast, cut corners short and don't run into walls. There aren't many speedstrats. Still, here's the ones I know about.

Cutscene skip: Foal and horses are rescued before the cutscene starts playing. Meaning that you can pause and exit the stage during the last cutscene of each trail.

Death abuse: In mountain trail 2, when the trail splits into two paths, both indicated by a sign, go left. You don't even need to feed the nearby raccoon. Once you rescued the foal on the broken bridge, take an intentionnal death to said racoon as this will send you back to the intersection.

Fast Disc Speed: This one is a given. Barbie Horse Adventure: Wild Horse Rescue is a CD game, not a DVD one, meaning that you can activate Fast Disc Speed in your PS2 settings before booting up the game for much, much faster load times.