Oh, this application is just begging to be done on a microcontroller. You don't have to build very much as there are so many microcontroller development boards already made and available for reasonable cost. I'm sure you can start with one of those and perhaps add simple driver transistors for the LEDs or maybe nothing at all. I can assure you that once you have the LEDs wired in to the microcontroller you will get a huge kick out of making them turn on and off as you desire. Great fun!
OK, lets see, you need one switch input for toggling between red and blue state, one reset switch (hidden button) and that's all the game inputs? For outputs, you will have one for RED, one for BLUE. That's not very many inputs and outputs so just about any micrcontroller will work, however I will suggest my own preference. Now, how about programming? Do you know any programming languages? I have used a compiler that is well suited to a novice programmer called BASCOM-AVR. It is a Basic language compiler that specifically works with the AVR family of micros, and it is available for free via download. It is easy to do a quick little project like this in BASCOM, especially if you have learned any programming languages at all. There are many other alternative platforms including PIC/C compiler and Arduino for example. There are many makers of development boards. For example:
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The development environment to write software for the AVR is all free. You can use BASCOM, or you can work in assembler with Atmels AVR Studio4, or you can work in C using one of several free systems. One thing you would need is a programmer, which is a small device that goes between you computer and your micro circuit board. If you are strapped for cash, you can make your own parallel port programmer for very little money. If you can afford one, the Atmel AVRISP-2 is a good programmer for around $40 or so. I have used both of these types.