Sounds like you are trying to learn electronics on your own without anyone to help and guide you.   Thats how I learned it too.  Electronics is only difficult to learn because there are quite a few "simple" things you must learn... and then  it starts to make sense.  Here are some of them that you must master before it all makes sense:  Ohms law.  prefixes (such as milli, micro, etc), using a multimeter, soldering, various components (diode,resistors,etc), how to read schematics,  understanding shorts & opens,  series/parallel, power supplies,  and a couple others i cant think of right now.  now, lets answer a couple of your questions:

MEASURING CURRENT:  yes, you must OPEN some portion of the circuit to measure current (amperage).  HOWEVER i dont recommend someone just learning do this.  If you do this WRONG you could damage your multimeter or damage the item you are testing.  For example, if the meter probes were improperly connected to GROUND while meter is set for AMPS.... you would damage the circuit under test.  so AMPERE readings are not for beginners.
do a lot of VOLTAGE and RESISTANCE testing before you tackle AMPS.  You should NOT cut an "arbitrary trace" unless you KNOW what kind of signals
and voltage are on that trace.... could be AC .DC .. could have RF on it, or it
could have  combinations of those on it.

CUTTING A TRACE:  when working on equipment for repairs... a technician would NOT CUT A TRACE unless no other method could be found.  You might be able to LIFT one leg of a component... or measure somewhere else more convenient... also, technicians RARELY do amperage measurements... we do VOLTAGE measurements (with circuit ON) and we do OHMS and DIODE JUNCTION tests (with circuit powered OFF).  amperage only rarely.  Cutting a trace causes more work to repair,  Easier to just remove one wire of the motor, if you want to measure amp draw of the motor.  Hope that helps you  a little. 

Here is a great site with some EXCELLENT downloadable electrical text books...

They cover a real lot of stuff, are easy to read and even I can understand them.

http://www.allaboutcircuits.com/

... how do you measure current in the arbitrary track on the PCB? Are you supposed to cut the track open, just to take one measurement?

There are various methods. But that depends on nature of the signals you want to measure.
If it's DC, you could measure the voltage at the terminals of a resistor on the track you want to know the current. Then, knowing the value of the resistor and the voltage at its terminals, you could calculate the current thanks to the Ohm's law. But this only works if the signal in the track is DC and constant.

Why there is no current flowing when resistor is connected and there is current when motor is in ?

If everything works correctly, there should be a tiny current of 1.7 mA. ( 3/1800 = 1.66 x 10^-3)
But maybe you can't see it because your multimeter is not enough sensitive or is set on the wrong decade.