different outputs based on button pushes?

The easiest way to do that would be to use a programmable microcontroller.  They greatly simplify time delays and you can do everything you want with one 8 pin chip.  Of course the catch is you have to be able to write a program, download it, and then test it.

If you go with a brand like Atmel AVR you can get the programming software for free, you can build a programmer for less then $3, and you can buy the chip itself for $1.50.  There are great intro to AVR programming sites on the net and this would be a perfect "first" project.

Radioshack doesn't sell Atmel parts - you'll need to order them from Mouser or Digikey or some other parts place.

To write your program you'll need to download AVRStudio from www.atmel.com.  You can then download a program called PonyProg ( http://www.lancos.com/prog.html) to download the .hex file you create to a chip through a parallel port (assuming you have a parallel port).  I built a parallel programmer years ago using only a couple of resistors and it worked great - I'm sure if you are interested you can find the circuit somewhere.

This is a great page to get the programming basics: http://www.avr-asm-tutorial.net/avr_en/beginner/index.html   The program you will need is pretty basic and shouldn't be difficult to write as long as you have some time and patience.

As far as hardware, I would use a Tiny12 or Tiny13.  Both are 8 pin ICs that are more then capable of doing what you want.

If you are interested and need help with the software let me know - it shouldn't take me more then 10 mintues to write something that works.