Is there a way for me to make chat invisible until a button is pressed?

Roblox has a service called StarterGui which you can use the function :SetCoreGuiEnabled on. This will then give you the option to set specific parts of the core gui.

http://wiki.roblox.com/index.php?title=Disabling_parts_of_the_game_interface

Using this, we can create a module in our script to have functions for disabling/enabling parts of the interface:

local hud = {}

-- hud module
-- By Scarious
print("Loading hud module")
do
	local sGui = game:GetService("StarterGui")
	
	function hud:enableLeaderboard(on)
		sGui:SetCoreGuiEnabled(Enum.CoreGuiType.PlayerList, on)
	end
	
	function hud:enableHealth(on)
		sGui:SetCoreGuiEnabled(Enum.CoreGuiType.Health, on)
	end
	
	function hud:enableBackpack(on)
		sGui:SetCoreGuiEnabled(Enum.CoreGuiType.Backpack, on)
	end
	
	function hud:enableChat(on)
		sGui:SetCoreGuiEnabled(Enum.CoreGuiType.Chat, on)
	end
	
	function hud:enableAll(on)
		sGui:SetCoreGuiEnabled(Enum.CoreGuiType.All, on)
	end
end

Now, 5 functions have been added:

-- hud:enableLeaderboard(true/false)
-- hud:enableHealth(true/false)
-- hud:enableBackpack(true/false)
-- hud:enableChat(true/false)
-- hud:enableAll(true/false)

for the purposes of this, we are interested with:

-- hud:enableLeaderboard(true/false)
-- hud:enableChat(true/false)

so, to disable the leaderboard and the chat at start we will use this:

local player = game.Players.LocalPlayer

function characterAdded()
	hud:enableChat(false)
	hud:enableLeaderboard(false)
end)

characterAdded()
player.CharacterAdded:Connect(characterAdded)

to re-enable it at any point, we will do as follows:

-- function for player clicking a button to enable leaderboard
hud:enableLeaderboard(true)

-- function for player clicking a button to enable chat
hud:enableChat(true)

That's about it, if you have any questions, feel free to ask. Note that this must be used in a local script (duh)