sequential leds using op amp
The answer given by @AaronD is pointing completely in the right direction, but it's leading you further than you need to go (the combinational logic), and it's not entirely telling you how to get there.

The suggested article on Rod Elliot's website (sound-au.com) is a great starting point, but doesn't quite cover, or truly explain what you need.

Pinching the core of 'Project 144' - although it's a well established principle - is considering a resistor ladder. It's a string of resistors in series, with one end connected to a voltage source (we'll call it Vin) and the opposite end connected to ground. Each junction between any two resistors will, by Ohm's law, be at a specific voltage between Vin and ground. I'm not going to explain the maths behind it - that's your job to find out.

I assume you know that op-amps can be used as comparators - again I'm not going to explain this. You connect each junction of the resistor ladder to the reference pin of a single comparator, and a voltage between Vin and ground (let's call this Vtr) to all of the comparators. As you should have figured out by now, as you increase/decrease Vtr the comparators will switch state one by one.

Thus:
1) You need to create a voltage source (Vtr) that increases or decreases with time, with a fairly linear response for 9 seconds - that's how long your outputs need to run for. A simple RC circuit will do. It's your choice which 'direction' you choose, although if I were doing it I'd pick a decreasing voltage; the rest of my answer assumes this. 

2) You need to build a resistor ladder, connected to your comparators. How many op-amps do you need? Well, you have a green light, a red light, and a buzzer; so three. That means you need four resistors. You need to pick values of resistors such that the junction between Ra (the topmost) and Rb (the second one) is the same voltage that your Vtr will have reach after your '2-4 seconds' required for the green LED to be lit. The next junction between Rb and Rc has to be the same voltage that Vtr will be after another '2-4 seconds' to cover the red LED. Guess what's next? Yes, the buzzer.







simulate this circuit &ndash; Schematic created using CircuitLab

The above schematic explains a little better the two points. This circuit is for a decreasing Vtr; if you want an increasing Vtr you'll have to change the order and orientation of the op-amps.

But what do you think happens when you connect the LEDs and the buzzer?
You should notice that when Vtr is equal to Vin, ALL the outputs are on! Of course they are, because all the comparators are being triggered. If you can't use anything other than op-amps then you can't use logic gates.

I don't know if you'll find this answer by Googling it, so I suppose it's not against the 'Stack Exchange will not do your homework' policy to give you the answer straight. Set the outputs of the comparators like this:







simulate this circuit

NOTE: I haven't included any dropping resistors in the above schematic - again that's up to you to calculate and include.

You should be able to work out what's going on and how/why it works. If you get a bit stuck, draw a truth table showing each comparator output ('hi' or 'lo') against time:

|  Time  |  Vsup  |  Comp1  |  Comp2  |  Comp3  |
=================================================
|  0-3   |   HI   |   LO    |   LO    |   LO    |    &lt;-- GREEN ON
|  3-6   |    ?   |    ?    |    ?    |    ?    |    &lt;-- RED ON
|  6-7.5 |    ?   |    ?    |    ?    |    ?    |    &lt;-- BUZZER ON
|  7.5+  |    ?   |    ?    |    ?    |    ?    |    &lt;-- ALL OFF


The 'Time' column is of course not precise, it's just an indication of the four stages of output. Fill in the gaps and you'll see why only one output is active at a time.

If you have questions please comment, but make sure you show that you've put some effort in to get further than you are now.