Think of a transistor as a diode with three pins for testing purposes. On all transistors, the arrows on their data sheet will indicate that electricity should only flow in that direction, identical to a diode. With your meter in diode test mode, put the black probe on the base pin 1 (left) and the red probe on the emitter pin 3 (right). If you see zero resistance on the meter, the transistor has shorted. If that checks out okay, put the red probe on pin 2 (middle), and the black probe on pin 3 (right). Again, if you see zero resistance, it is shorted. You should see infinite resistance in both of those tests, just like a diode would if you try to go in the opposite direction.
That diagram may look a little confusing, to explain better, this particular transistor is called a Darlington pair. It is just two transistors sandwiched together in one little package.
Keep in mind those instructions are specific to testing TIP102/122/36 transistors. The pre-drivers and other transistors on your machine have a different pin-out, but the testing and information I gave you still applies.
The blue box thing is most likely a relay. Take a picture of the board and upload it so we can help you better.