How to make a player face a certain direction when they teleport?

Using MoveTo isn't the way to do this, all parts will move with the hrp, so just rotate/move the hrp to teleport a player

(HRP = HumanoidRootPart)

For this, use CFrame

Btw, use remote events, don't teleport client side

Player = game.Players.LocalPlayer
SpawnTo = game.Workspace.SpawnTele
TeleportButton = script.Parent.TeleportButton
HasPressedTeleport = false


function TeleportPlayer()
    if HasPressedTeleport == false then
        local char = Player.Character

	if char then -- Might not be loaded
		local HRP = char:WaitForChild('HumanoidRootPart')
		HRP.CFrame = SpawnTo.CFrame -- You can rotate 'spawnto' to change the players rotation or use cframe angles
	end
    end
end

TeleportButton.MouseButton1Down:connect(TeleportPlayer)

If you have any question, or I made a typo/mistake, just ask :)

So as an edit for your question, the best way to do it is to set the camera quickly to attach, then back to custom

Player = game.Players.LocalPlayer
SpawnTo = game.Workspace.SpawnTele
TeleportButton = script.Parent.TeleportButton
HasPressedTeleport = false
local Cam = workspace.CurrentCamera


function TeleportPlayer()
    if HasPressedTeleport == false then
        local char = Player.Character

	if char then -- Might not be loaded
		local HRP = char:WaitForChild('HumanoidRootPart')
		HRP.CFrame = SpawnTo.CFrame -- You can rotate 'spawnto' to change the players rotation or use cframe angles

		repeat
			wait()
			Cam.CameraType = Enum.CameraType.Attach
		until Cam.CameraType == Enum.CameraType.Attach

		repeat
			wait()
			Cam.CameraType = Enum.CameraType.Custom
		until Cam.CameraType == Enum.CameraType.Custom
	end		
    end
end

TeleportButton.MouseButton1Down:connect(TeleportPlayer)