Walking sideways/backwards slows you down. Exception is Jack, he always moves at the same speed.
Shooting your weapon slows you down. Exception here is Jack again (both the repulsor and the pistols) and some others, like Caira’s granade launcher or Tech Sgt. Hank’s shield charger.
Hovering with you jetpack does NOT make you faster. If you use your jetpack, double tap to get the boost.
Climbing walls is situational. Like, when you are on the chase you can double tap to reach the top faster. If you are in combat or expect an ambush, you should save jetpack and use the normal climb.
When kiting as Caira, definitely run forwards, not backwards. Running around trees/rocks/pipes/something will help you block monster abilities like a rock throw. Pay attention though to not break line of sight to your support if it’s Hank or Sunny, or to your Trapper if it’s someone like Jack or Griffin. If there is an open area and nothing for you to kite around, run in zigzag patterns and stay relatively close to your assault, so that he can do some damage.
I hope you got a good headset, because you need it when you play as Caira ^^ When you look at the ground most of the time, you can still hear where the monster is. You can hear when he is using abilities, so you can dodge them. And you can hear when he moves away from you, so you can look up and heal your teammates.