Hey, welcome to the addiction. :-) Sorry, this is kind of a late reply.
Download the Arduino software if you haven't yet. Get used to the IDE, and check out the example sketches. Don't aim too high at first, or else you'll most likely get frustrated. Your first few experiments should be on the order of getting an LED to blink.
There are dozens of great first-day-with-Arduino tutorials online. I recommend the Adafruit tutorials. Have you done any programming before? How are you with electronics?
I would guess your device #2 is an ICSP programmer, but I would need more info to be sure. ICSP is used to program raw chips if you ever decide to breadboard unprogrammed AVR microprocessors (e.g., taking the CPU chip off the Uno board, that is.) That's a little more advanced, and not strictly necessary since the Uno board gives you a way to talk to the AVR chip via USB. But, if you get into this and start making your own standalone circuits, you'll use it a lot.
BTW, in case you're not aware already, Atmel makes the AVR line of microprocessors. "AVR" is the family, kind of like an Intel Core i7 is an "x86" family CPU. There are 8-bit and 32-bit AVRs, just like there are 32-bit (x86) and 64-bit (x64 or x86-64) Intel and AMD CPUs. In the 8-bit AVR family, there are ATtiny and ATmega lines. The Tinys are smaller and more limited, but still fairly powerful. The Megas have more hardware features -- I/O pins, hardware serial UARTs, more memory, etc. They're all based on a similar core architecture and share more similarities than differences.
Arduino is a project to provide a starter kit environment of hardware and software to make AVR programming easier. It's based on C/C++, uses completely standard syntax, but hides some of the housekeeping chores from you so you can focus on rapid development. It's very powerful, while still being more user-friendly than traditional microprocessor development. But, if you ever want to get closer to the hardware, you can download AVR Studio and code in plain vanilla C, C++, or ASM as well, without everything you've learned being wasted effort.
As for hardware, the Uno uses the ATmega 328P chip specifically, which is one of the middle-range 8-bit AVRs. Other common Arduino-land micros are (from least powerful to most) the ATmega 8A, ATmega 168, and ATmega 2560.
The H-bridge is your motor driver. The AVR chip itself is not capable of high-current output, so connecting a motor directly to the output pins will lead to broken hardware and sadness. The H-bridge takes low-level control signals from the Uno and drives your power-hungry devices off a more capable power supply.
So that should clear up some initial stuff. Feel free to ask when you have more specific questions. This is a very friendly and helpful forum, so don't feel under-qualified to be here. As long as you make an honest effort, the members here will go to lengths to help you out.