Yes, your circuit should be simple enough to implement with timers, discrete logic, and some small IC counters and flip flops. If you are fairly neat in your workmanship you can probably even get away with building the circuit on a breadboard and then fabricate a simple circuit board. To me this sounds like a good fun project that would be challenging but do-able.
How familiar are you with truth tables and minimization through Karnaugh maps? What about state machine diagrams? If you aren't very familiar with this, I would recommend that you get a book on digital logic and switching. Probably about the time you get half way through the book you will have everything you need to do this project. A book on digital electronics geared towards hobby work would likely even have projects to build a bread board system. I have a book on my shelf "Digital Electronics Guidebook", by Mike Predko, which I believe I ordered through Amazon. This type of book may be a good choice for you. It covers the digital and analog electronics including transistors, counters, flip flops, 555s and up to simple processor based stuff.
With the bread board approach, you can then build up a small section, such as one of the truth tables or state machines, counters, etc and experiment with each one seperately and then start putting them together. If you don't have one, you can probably get a bread board at radio shack or an online company. I had one floating around but I can't find it to tell you where I got it, but I think it was something along the lines of "circuit specialists".
The way I would recommend starting is to think about the inputs you need / have and the outputs you want. Then either draw a set of truth tables and or state transition diagrams. This will give you the basic logic equations, which you can the minimize to reduce the number of gates, wiring., and cost.
If you are ambitious enough, this would be a good PIC project. I have seen books on PICs even in places like Barnes and Nobles and Borders. I say that this would be a good PIC project because a lot of the functionality you are after would be pretty easy to implement with simple software. You could then use the IO pins of the chip to drive the LEDs or drive some decoders, etc.