Fighting Fetches can be quite frustrating, because they run faster than Red can and may inevitably catch up to her, causing lots of damage up close. Jaunting away and fighting them with projectile attacks like Breach or Bounce is a good strategy. Remember, the Fetch has no ranged attacks of any kind, so it must close the distance before it can damage you. Often, the best way to deal with a Fetch is to fire a large number of Breach or Bounce shots before it even starts to chase you. Get, on the other hand, is a bad choice, because closing the gap between the two of you is exactly what the Fetch wants.
In version 2.0, the Fetch gains a hunting cloak, allowing it to Mask itself. Plan your attacks carefully, because the Fetch cannot be hit while it is invisible, and it will also get an Unmask bonus for attacking you.
In version 3.0, the Fetch gains a stunning bark, which will paralyze Red briefly and open her up for even more melee damage.