How do I adjust the brightness of a 3 V bulb?
There are two basic ways to control the brightness.



Analog(ue): 

Something that will continuously control and change the current through the bulb. This could be just a variable resistor. In the circuit shown it is a BJT (transistor). 

The variable resistor (VR1) is used as a potential divider and so the voltage at the base of the BJT can be altered between +V and 0V. R1 prevents too much current going into the base of the BJT when VR1 wiper connects to the +V rail. The transistor has current amplification so the variable resistor does not have to take the current through the bulb. 

Note I've set the supply voltage slightly higher than 3V so full brightness can be obtained when VR1 is slightly below maximum. This also allows for the voltage drop across the transistor. 

Digital.

The bulb is turned ON and OFF very rapidly so it doesn't flicker. The circuit (in this case a 555 timer) can be adjusted to vary the ON and OFF (mark/space ratio) time of the pulse. At 50-50 (equal ON,OFF times), mid setting of the variable resistor, the POWER is half the maximum. At 25-75 it is less ON than it is OFF so the bulb is dimmer. At 75-25 the bulb is more ON than OFF so it is brighter. The circuit doesn't produce a 100 -0 or a 0-100 setting but should get close enough to control a full range of brightness.

Q1 is used to switch the current through the bulb ON and OFF. D3 ensures the output from the 555 when it is LOW (it doesn't get to 0V) doesn't keep the transistor ON. R2 limits the current into the base of the BJT.

This type of circuit (PWM or pulse width modulation) is commonly used to control power to motors, transformers etc.