How to send messages to the chat with a script?

In terms of your original question, you want to use the SetCore function in StarterGui (see here) for an example of this. You will need to run the script in a LocalScript for it to work, and the message will only display on the player's screen for which the script is running.

Making a custom chat script is quite a different topic and it's not something I can explain into an answer to this question, but I suggest you check out this wiki article for more information. Sorry I can't explain this in an answer, but it is a very complicated topic and it is very technical.

If you want the server to "Chat" when the player joins and the player leaves, you would need to put this into a LocalScript inside StarterPlayerScript:

See comments in the code to get a better insight of how this script works!

If there are any typos, then please excuse me, I wrote this code here and now.

pcall(function() --Event that should fire when the player joins
    game.StarterGui:SetCore("ChatMakeSystemMessage", {
        Text = "[Server]: " ..game.Players.LocalPlayer.Name.. " has joined the game!";
        Color = Color3.new(255, 255, 255); --The color of the text
        Font = Enum.Font.SourceSansBold; --The font of the text
        FontSize = Enum.FontSize.Size24; --The size of the text
    })
end)

game.Players.ChildAdded:connect(function(player) --Event that fires when player joined
    if not pcall(function() --If the pcall() function was not called
        game.StarterGui:SetCore("ChatMakeSystemMessage", { --Gets the chat
            Text = "[Server]:  " ..player.Name .. " has joined the game!"); --Sets the text
            Color = Color3.new(255,255,255); --Sets the color of the text
            Font = Enum.Font.SourceSansBold; --Sets the font of the text
            FontSize = Enum.FontZise.Size24; --Sets the size of the text
        })
    end)      then
        print("Error") --prints error if the pcall() function was not called
    end
end)

game.Players.ChildRemoved:connect(function(player) --Event that fires when player left
    if not pcall(function() --if the pcall() function was NOT called
        game.StarterGui:SetCore("ChatMakeSystemMessage", { --Gets the chat
            Text = "[Server]:  " ..player.Name .. " has left the game!"); --Sets the text
            Color = Color3.new(255,255,255); --Sets the color of the text
            Font = Enum.Font.SourceSansBold; --Sets the font of the text
            FontSize = Enum.FontZise.Size24; --Sets the size of the text
        })
    end)      then
        print("Error") --prints error if the pcall() function was not called
    end
end)

For more information, visit this link: http://wiki.roblox.com/index.php?title=API:Class/StarterGui/SetCore

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