How can i make player move forward when clicked on a button like R?

In the example you used its obvious that you tried to use the character animations, I will tell you this now, this will NEVER work, animations will only move character parts/joints never the HumanoidRootPart (Which is where the camera is centered on). You can fix all this by using a body velocity. (Also make sure to have the animation not move the character or else it will do both movements at the same time). Makes sure this is all a local script.

local player = game:GetService("Players").LocalPlayer
local Character = player.Character or player.CharacterAdded:Wait()
local RootPart = Character.HumanoidRootPart
-- if you already have the variables above in your own script there is no need to write them again

--instead of using animations we will use a bodyvelocity, it is basically speed in a given direction

local BV = Instance.new("BodyVelocity")
BV.MaxForce = Vector3.new(math.huge,math.huge,math.huge)
--in order to make the character actually move, we will give them speed in the direction they are facing
BV.Velocity = RootPart.CFrame.lookVector.Unit * 50 -- you can change this if its too fast or too slow
--make sure it is parented to the "HumanoidRootPart" if it's not the script won't work.
BV.Parent = RootPart
wait(2) -- or however long you want the speed to last
BV:Destroy()

I hope this fixed your problem and please mark this as the answer if so! :)