There is no straight forward answer to your base resistor question. It all depends how the rest of the circuit is set up. If you find a base resistor in somebody else's circuit you can safely assume it is there for a good reason.
Try leaving it out and see what happend ( might cost you a transistor, depending on what you feed into the base :wink: )
There should be some good transistor tutorials on the net. Try Google to find some.
About your current question, current flows THROUGH a component. You mesure it by breaking the connection to the component and inserting an Ampmeter (I=V/R).
As the current flows through a resistor it sets up a voltage drop ACROSS it, depending on theresistor value and the current (V=IxR).
You measure this voltage by placing the voltmeter probes to each end of the resistor.
For a current to flow you have to have a complete path, from positive through various components to negative. If there is more than one path the current splits, the ratio depending on the individual resistances. The total current flowing INTO a circuit always equals the total current flowing OUT of it.
The various components in a circuit develop individual voltage drops across them as current flows.