Tamed pet AI for pathfinding is rather awful. They get stuck where you would never expect. You can try "All Come" for better pathfinding, but the pet will stop following when it gets to you and you need to say "All Follow" again.
As far as speed, it will get faster as the Dex trains up to 125 max for most pets. At that dex/speed, it should be able to keep up with you at a run on foot, but not mounted.