The Arduino is really popular, and it is open-source hardware, so there are variations, including the third-party Freeduinos called Bare Bones Board, the Really Bare Bones Board, and the Boarduino, which are very nice for using with breadboards. The standard Arduino is adapted using things called Shields. You can make your own breadboard shield or buy something like this ProtoShield Kit.
The Arduino uses an FTDI USB-RS232 chip or cable. [The standard Arduino has it build in, while the most of the bread board ones use a special cable that has the chip inside, and saves you some money.] There is a built-in driver for Linux, and Mac OS X and Windows users can download a driver. Effectively, this means that the Arduino acts as a device communicating on a serial port, and so it is very easy to communicate with.
To use it, you also download software. It comes with an IDE, an AVR-GCC toolchain, a really nice library, and software to upload your program on to the chip. It hides most of the details from you, and has a great community. It is programmed in a language called "Wiring", but it is really C++.
Lastly, being open-source hardware, there are schematics out there. Indeed, I seem to recall reading that it was fairly easy to build an earlier model. Searching for "Arduino schematics" in your favourite search engine will give you good results.
As for kits, the Maker Shed offers a lot of Arduino-related items in stock. They appear to ship around the world. Some ones of interest include:
The Shoppe at Wulfden (USA) has a nice experimenters kits, and appears to ship internationally.
Solarbotics (Canada) has a ARDX Arduino Experimenter's Kit, a Freeduino Starter Bundle - Ultimate and an Arduino Starter Bundle - Basic (and Ultimate), and appears to ship internationally.
The main Arduino's "buy" page lists Arduino vendors in all areas of the globe.
Other places I would check include Adafruit Industries and Sparkfun Electronics (both in the US).
Best Answer
Generally, you won't tend to find any DIP-packaged USB controllers or USB mass storage-grade parts because of the speed of operation. I'm assuming you're talking about a breadboard like this:
This is because the USB 2.0 protocol uses a SerDes differential line that runs effectively at 480 MHz. Now, underside the breadboard, we're looking at something different:
These parallel strips of metal are what help to retain pins, but they have a serious side-effect. They raise the capacitance of any circuits put on them severely. Because capacitors transmit AC and serve as a short to DC, your USB signal could effectively bleed through the breadboard at will. There are other effects, but in practice, standard prototyping breadboards of this style of construction are limited to a maximum speed of 10 MHz (significantly less for analog circuits) or so before that capacitance really starts messing with a circuit too much.
If you're still interested in learning about USB, there are multiple avenues you could still pursue. You could get a Arduino/Teensy/ChipKit/etc. board and find a USB stack supporting the Mass Storage Class (for example, this one for PIC-based microcontrollers) and throw on a couple SPI flashes and use that to experiment with how USB handles mass storage data. You can always move up to the "big league" of actual USB mass storage device creation when you have the skill set or the want to do so.