A digital approach could be done using a CD4060 to provide a stable clock at a specific rate. The output of the clock would feed a CD4017 which will count the clock and decode the count to one of ten outputs. This provides the sequencing. Connect the first three outputs to FET switches to drive the lamps, and add a suitable reset circuit that resets on power-up, starts counting when a button is pressed, and stops the sequence after three lamps have lit.
Using CMOS logic, such as 4000 series, allows the use of direct battery voltage for power supply (15V maximum, so perhaps add a zener for protection). The circuit would, therefore, include two integrated circuits (costing $0.53 plus $0.59), several resistors and capacitors, three power FETS, a scrap of perf board to wire it together on, and the wire to hook it all up with.
The circuit is digital and therefore highly predictable and stable. You can adjust the speed of the lamp sequence by varying one resistor. I like to use the 4060 as a time base for slow outputs because you can run the oscillator at a higher frequency, like 16384 Hz, and use a smaller value polystyrene, polyester or similar plastic capacitor to determine the frequency of the oscillator. These types of capacitors are extremely stable with temperature variations and so are a much better choice for timing applications, but they are most readily available in smaller values, like 0.01 uF for example.
Of course, if you have a 555 or other timing IC handy, you can use that instead of a 4060 to give you a 1 second clock.
I think this is a relatively simple and straightforward circuit to build. If you would like a schematic I can draw one up and post it for you, but I would need to know what wattage of lamps you intend to use and whether you intend to have a "red light" function. The red light function would need a startline light beam sensor and makes things a fair bit more complicated.