You can use and IC called a 4017. It has 10 outputs that turn on one after the other. If you connect the 7th output to the reset input, then it will cycle round after the 6th step.
You need something to time it.
Electronoize Playshop: 4093 Oscillator shows a simple oscillator.
**broken link removed** shows the idea very well.
You need a transistor for each output of the 4017 as the outputs can't provide enough current unaided.
It would be a good idea to have just two or three LEDs in series. If you have two in series, that will have a maximum voltage of 6 V. You then need a resistor of (13.8 - 6) / 0.04 = 195 ohms in series with each pair. 220 ohms is a standard resistor value, so you could use that an just get a bit less current. You can have as many groups, each with two LEDs and a 220 ohm resistor in series, by wiring each group in parallel.
There is an online calculator **broken link removed** here. However, it doesn't take any account of the variation in LED voltages and power supply voltage. For a car, where the supply voltage varies, the total of the forward voltages of the LEDs shouldn't be more than about 8 V. That way the resistor voltage will change from about 12 - 8 = 4 V to 13.8 - 8 = 5.8 V. That means that the resistor current will change by a factor of 5.8 / 4 = 1.45 and the LED current is the same so changes by the same ratio. If you have more LEDs in series, so more voltage you get less voltage on the resistor so the current changes by a bigger factor.
For wiring, I suggest using stripboard.
Stripboard - Wikipedia, the free encyclopedia