OK, I think a good place to start would be to make a provisional choice of microcontroller - at least the architecture. Microchip PICs are very popular, are fairly easy to program in C, and have plenty of examples and tutorials. Many people here have experience with them, including me, Steve, Gryd3, BobK, and others, who can advise you on the hardware and firmware aspects. You will have to take quite a bit of the initiative yourself though.
Gryd3's suggestion of starting with a pre-built prototyping/evaluation board with an MCU and programmer is a very good one. Once you get the code working you can do your own stripboard circuit or PCB. Here are four PIC evaluation packages from Microchip available through Digi-Key:
http://www.digikey.com/product-detail/en/Q7442559/LPC2-PROMO-ND/3880043
USD 34. Includes a number of blank PIC devices but no programmer/debugger device.
http://www.digikey.com/product-detail/en/Q7442546/LPC1-PROMO-ND/3880042
USD 68. Includes a number of PIC MCUs (three 8-pin devices and two 14-pin devices) and a PICkit 3 programmer/debugger module, which you will need - probably your best option.
http://www.digikey.com/product-detail/en/DV164130/DV164130-ND/3671621
USD 72. Includes a PICkit 3 programmer/debugger module, which you will need, but it seems, no PIC MCUs.
http://www.digikey.com/product-detail/en/DM163022-1/DM163022-1-ND/2696124
USD 120. Evaluation board with an LCD. Includes some blank MCUs but no programmer/debugger.
This application needs no special features and very little processing power. The choice of MCU is really driven by how much I/O you need. Each MCU needs two pins for VDD and VSS, and two pins for the crystal. That leaves either 4 (for an 8-pin PIC) or 10 (for a 14-pin PIC) more pins for general I/O; one of these is only able to be an input to the device, but the others can be either inputs or outputs.
That's fine for this case, because you're going to have a pushbutton. And if you only want three LEDs, an 8-pin device is suitable, but you have no expansion opportunities. Actually you do, because three lines are enough to drive external LED driver ICs (there are several options here), but it's simpler to drive the LEDs with a 1:1 I/O mapping. Something for you to consider at a later stage.
PIC microcontrollers can be programmed in several languages. The commonest are assembly language and C. C is quicker to learn and there is a lot of sample C code for PICs. C is actually a general purpose programming language that can also be used to program PC-type computers; microcontrollers are very resource-constrained compared to PCs so you should look at sample code that is specifically written for microcontrollers.
The development environment is called MPLAB and it's a free download from microchip.com. There are two generations of MPLAB: MPLAB 8 point something, the last version of the old architecture, and MPLAB X (the 'X' means ten, like Mac OS X) which is the way of the future. Some have criticised MPLAB X but I haven't had any big problems with it. It includes various sample programs, and there's probably code there specifically written to run on those evaluation boards.
MPLAB programs PIC devices via the PICkit 2 or PICkit 3 programmer, which connects via USB. Like MPLAB, there is some argument about PICkit 2 vs. PICkit 3. PICkit 2 has some extra features, which probably won't be important for your application. PICkit 3 is supported for current and future devices. Your choice.
The out-of-box experience should be reasonably painless, and you then have a system that you can muck around with to your heart's content. Make a little change, recompile and reprogram, and see what happens. Perfect for a hands-on person. A lot of fun too. Then you can start thinking about the specific hardware and firmware requirements for your project.
For my simple ready, set, go project, I don't need arduino. The microcontroller should be adequate.
All the components are foreign to me together with engineering that goes into the project.
If someone were to give me the parts list and wiring diagram, I would be a happy camper. I cannot find a ready made component on the market that will handle my needs.
If I could buy the necessary parts on Monday, complete and test the assembly on Tuesday, I could ship out two units on Wednesday and make enough money to by gasoline for a week. I agree that the whole thing is relatively simple, but way past my education for this subject.
I'll keep the notes you sent me. Many thanks. I'm a poor old man struggling to make ends meet grasping at ideas that seem to be doable. The subject material that you address include a large vocabulary with functionality that I am slow to grasp. Even the 555 chip has so many variables that interact with each other, I cannot make working sense of it. At this point, I'm not interested in possibilities, but a specific final product I can sell to put bread on the table.
Michael