Does the PIC and the TINY use the same programming language?
Both are normally programmed in C or assembly language. Microchip's free programming environment for the PIC includes a C compiler and an assembler. I think Atmel's equivalent does too.
They are very resource-constrained devices and the C language implementations have a number of limitations but would be fine for what you want to do. Writing in assembly language requires more learning, and the PIC and AVR have different instruction sets.
Is there any advantage other than price?
Well, in my opinion the AVR is a much tidier architecture, but this is only really significant if you're programming in assembly language. If you're programming in C, the compilers take care of those details, so no. Both devices have similar built-in peripherals - reset on power-up and brown-out, simple timer, on-board clock oscillator, weak pullups.
PIC and AVR are both fairly widely used, but I think the PIC is more popular. I think they're a pretty poor architecture, so that's unfortunate, but they certainly are cheap!
Next up would programming. What would be the simplest, most cost effective way to connect & program.
Like chopnhack, I have used the PICkit 3 programming adapter, which plugs into a USB port and is supported by the MPLAB X development environment. It's a long time since I used the AVR, and back then I used a very simple programming interface using the PC's parallel port, but modern machines don't have parallel ports, and USB parallel ports don't work, and I don't know what Atmel's entry level programming system is now.
I'm new to this, but the little research I have done has shown me that what I am intending to do is very basic. Just define the input and output, then define input conditions that would force the output high or low. This will be a fun learning experience for me, but I don't want to tie up too much time doing it. I will be working with Windows7 laptops & desktops. Thanks again!
Yes, you're right. You'll need to program the important internal peripherals - set the brown-out voltage, enable the watchdog if you want to use it, configure the I/O directions and weak pullups, configure the timer, and structure your code so that it does the debouncing, timing and output control, and is tidy and reliable.
We have several members here, including me, who can advise you on how to implement your requirements, once you've made a start.