Hi Andi
If you post the circuit that you tried already we could suggest improvements. Alternatively, if I was starting from scratch, I would first figure out how to make the green one come on for 5 seconds then go out. Once you have this working, you can use the output of this circuit to trigger another circuit that makes the red one come on.
To make a green one come on for 5 seconds, you might consider using a timer IC like the 555. You would design one 555 to trigger from the toggle switch in monostable mode. The 555 output would drive the green LED and would also drive a transistor switch. The transistor switch would turn on the red LED when the green LED turns off.
I actually prefer to use the 4060 CMOS counter (or a 74HC4060) for this kind of application but its a bit more complicated to explain. You wire the clock terminals as an RC oscillator clock, and you choose the divider output that divides your clock frequency down to 0.1Hz and wire this divider output back to the reset input. You will need on additional flip flop ic that also toggles from this divider output. The toggle switch must be wired to this flip flop also so that the flip flop controls when the 4060 runs or when it resets. The green LED will be driven by one of the flip flop outputs, while the red LED is driven by the other flip flop output. Like I said, its hard to explain in words but simple in a schematic.