Shoot only 3-star animals - if you can get close to an individual of a species you can study it. After studying a species, the scent trails you can see in eagle eye will also tell you which species and what rating the creature is.
Use appropriate ammunition. For small creatures (rabbits and smaller), use either small creature arrows (can be crafted with raven feathers) or buy a varmint gun from the gunsmith. Normal arrows seem best for most deer (varmint gun needed to head shots to drop a pronghorn doe, and the pelt lost score, even though they were both headshots). The repeater for elk. Bison and Grizzlies need something more powerful for a single shot.
Skin the creature immediately. Small pelts can go in the satchel. Larger pelts can be stowed on the horse. An unskinned carcass will degrade over time, even when it is stowed on the horse, so don't carry the creature away and then skin it.