There are a few things you need to be aware of, concerning attaching electronics to littleBits: First and foremost, is the voltage MUST be a regulated 5 volts. That is, it should be 5 volts when the circuit is open and 5 volts when drawing the maximum current (which has a limit, but I can't remember what it is, but the cloud bit draws 2 amps.) You will need to use the incoming power, and also pass it on to the next bit. It isn't very hard to do, but you need to be aware of it.
Second, the littleBits system has 3 and only 3 conductors. They are, from the bottom pin of the bitSnap connector, going up: 1) a ground (also known as 0 volts), 2) a a signal, which can be as low as 0 volts (but never negative), and as high as 5 volts (but no higher.) At the top of the bitSnap is the 3) 5 volt power line. That's about it. There are some other conventions, but as long as you respect the voltages, and keep all your data as an analog ttl (5 volt) signal, everything else is fair game.
When I don't want to use a proto bit to connect some circuit on a breadboard, I use these things I made by cutting a wire bit in half, and soldering pins onto the cut ends of the wires, Then I can stick the pins into the breadboard, and attach bits to the bitSnap connector on the wire: