Confusion over Transistors for a switching circuit with an Arduino Mega 2560
A transistor amplifies current, not voltage. 

The first thing to check is the current of your LEDs. In your current configuration, it is between 25 and 30 mA. This is too high for most LEDs. Check the datasheet of your LEDs for the actual forward voltage, and current rating.
To calculate the value for the resistor in your circuit, use Kirchhoff's and Ohm's laws. According to Kirchhoff: VResistor + 3 VLED = 12 V. Let's assume that the LED's forward voltage drop VLED = 1.8 V. This means that VResistor = 12 V - 3 VLED = 6.6 V. Ohm's law will then tell you the relationship between the voltage across the resistor and the current through it: R = VResistor / I. The current through the resistor is the same as the current through the LEDs. Let's assume that this is 20 mA. Therefore: R = VResistor / I = 6.6 V / 0.020 A = 330 Ω.

The next thing to calculate is the resistor between the Arduino and the base of the transistor. As mentioned before: the voltage on the base does not matter, the current does. You want to use the transistor as a switch, so you have to drive it into saturation. As a general rule of thumb, you want the base current to be a tenth of the collector current in that case: IBase = ICollector / 10. The current through the collector is the same as the current through the LEDs, so IBase = 20 mA / 10 = 2 mA.  

In a similar way as before, we can calculate the base resistor: VResistor + VBase-Emitter = VArduino. The voltage of the Arduino pin is most likely 5 V, and the Base-Emitter voltage of a silicon transistor is around 700 mV (look it up in the datasheet): VResistor = 5 V - 0.7 V = 4.3 V. Then use Ohm's law again to calculate the resistor value: RBase = VResistor / IBase = 4.3 V / 0.002 A = 2150 Ω ≈ 2.2 kΩ.

The current has to flow through the base to the emitter. For current to flow, you need at least two connections: current flows from the Arduino pin, through the transistor, to ground. You have to connect the emitter of the transistor to the Arduino's ground.

You can use pretty much any normal transistor that has a maximum collector current that is larger than the LED current, and a maximum collector-emitter voltage greater than your power supply.