How do I made a gamemode selection screen?

If your game is FilteringEnabled, use a RemoteEvent. I don't know the layout you want it to be like. Let's say you have a frame holding all the TextButtons that the player is to press, we'll have to create a remoteevent.

--Server Code

local re = game.ReplicatedStorage.RemoteEvent
local gamemode

re.OnServerEvent:Connect(function(player,txt)
      gamemode = txt
      print(txt)
end)

--Client Code

local gmbuttonframe = script.Parent
local re = game.ReplicatedStorage.RemoteEvent

for _,v in pairs(gmbuttonframe:GetChildren()) do
      if v:IsA('TextButton') then
            v.MouseButton1Down:Connect(function()
                  re:FireServer(v.Text)
            end)
      end
end

Everytime, you click a button on the frame, the server sets the 'gamemode' variable to the text of the button clicked. If the player pressed a button that's text is 'Free - For - All', the server will print 'Free - For - All' in the output.

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