How would I make a part face the player for only them?

There is a server/client boundary that exists, and only the player can see the client boundary. Other players cannot see other players client boundaries.

The answer would be to use a LocalScript , which gives access to this client boundary.

First, we need to use CFrame. It's not complicated, and is very similiar to Vector3, but comes with a rotation matrix. More on the developer API above.

-- In localscript, running in PlayerScripts
local part = game.Workspace.Part -- Get the part from workspace
local target = game.Workspace.Target -- Get the target the part will be pointing at

part.CFrame = CFrame.new(part.Position,target.Position) -- Rotate the part

Now while this does solve your problem, you may want it to constantly point at the target while it moves. This can be solved by using RunService, which allows us to call a function every frame, much faster then any while wait() do.

-- In localscript, running in PlayerScripts
local part = game.Workspace.Part -- Get the part from workspace
local target = game.Workspace.Target -- Get the target the part will be pointing at

local runservice = game:GetService("RunService") -- Get runservice

runservice.RenderStepped:Connect(function() -- Define the function and attach it to RenderStepped
	part.CFrame = CFrame.new(part.Position,target.Position) -- Rotate the part
end)

This should solve your problem. If there are errors, please comment.