How to I run functions inside a Table?

Your first script segment can be evaluated like this:

Functions = {
[1] = Test("Hello!"),
}
--[1] is the same as this:
Functions = {Test("Hello!")}
--Test("Hello!") asks lua to run the function. Test calls 'print' but doesn't return anything, making this line evaluate to:
Functions = {}

Clearly this isn't what you want. If you want Functions[1] to print "Hello!", use an anonymous function:

Functions = {
    [1] = function() Test("Hello!") end,
}
--Usage:
Functions[1]()


--Or don't use an anonymous function:

Functions = {
    [1] = Test      --Notice how we don't call it here, we just assign it to index 1 of the table
}
--Usage:
Functions[1]("Hello!")


--Or use Functions like a dictionary:

Functions = {}
Functions.Test = Test
--Usage:
Functions.Test("Hello!")

--If Functions is just supposed to be a list of functions, you don't need the [1] syntax; just do:
Functions = {
    Test,
    Test2,
    function() print("Anonymous function") end,
    --etc
}
--Functions[1] is Test, Functions[2] is Test2, etc

I setup my functions inside tables like this

local table = {
    func = function()
        print("hey look at me")
    end
}

table.func()

wiki page with examples