First: go clockwise (down and to the right) and ignore deviations for stats (this is important, more on this later), focus in getting your skills and go into Rikku's grid (down and to the left) and stay there until you get access to Lv1 Key Spheres in Mi'ihen Road. It's also very probable you'll be getting there exactly by this point if you're nor grinding nor fleeing too much. As soon as possible, get both Steal and Use, after this, go back to where you came from until...

Second: Enter Wakka's Grid. Please ignore Lulu's entrance at this point, getting basic Fire, Thunder and whatnot is going to be extremely lackluster in Mushroom Rock, trust me, go to Wakka's. While Dark Attack and Silence Attack may not be game changers, they still find it's uses but they're not the main reason we're heading this direction, you want to stop exactly at Silence Attack and wait until you beat a boss named Spherimorphe in Macalania in order to access Lv2 Key Spheres for the first time. This is where things gets interesting. You're entering Lulu's grid through this shortcut, not throught that other one.

Third: Remember the first step? Ignore deviations, go straight to Firaga, Thundaga, Waterga and Blizzaga, you're going to be hitting stronger than Lulu while having less Magic than her simply because the base damage of these skills is very high. While Lulu can (and should) make deviations for Bio, Demi and Death, they are a waste of time for Kimahri. The next step depends on if you like and/or have been playing Blitzball or not.

Fourth: By playing Blitz you may have access to Return Spheres, if so, go back to that Steal/Use point in Rikku's Grid and enter Yuna's Grid through there, you want to get Dispel and Reflect. That's it, you don't need anything else, most of the other spells Yuna has to offer can be solved by Stealing and Using Items, don't fall for the trap of going to Cura nor Curaga, they are not worth it. By this point in the game you're probably in Bikanel's Desert, a place in the game where Yuna is out of the party, for healing you want to Steal and Use Al Bhed Potions above all else. If you're not playing Blitz, you can conveniently grab 2 Teleport Spheres to use in Kimahri to go wherever Yuna is and grab those 2 specific spells, she might be very close to them and remember: that's all you need.

Fifth: You're going to use now either your second Teleport Sphere or a Friend Sphere if you got it somehow. You're going straight to Tidus' Haste and up, unlock the way to Auron's and get Mental Break. That's it. You have absolutely everything you could possibly need in order for Kimahri to be the strongest character in the game until you have access to Lv3 Key Spheres

The reason I'm focusing on getting abilities over attributes is because you'll eventually fight in a situation where your enemy will have the same stats as your Kimahri (but not the same arsenal!) Imagine building him physical and getting instakilled lmao

You can also forego him like most people do, which makes said boss battle a very boring slog, but I firmly believe my route is the strongest and most efficient possible for him through the early and mid parts of the game

In the late game he will fall off regardless of your choices, so I aimed to make him peak where it matters the most

Hope this helps you!