To drive the 7-segment displays, I'd recommend either using transistors (there is a limit to the amount of current the ATMega can drive/sink) or another chip which will make driving it easier. Maybe look at an I2C port expander for driving the 7-segment displays. That way you can drive both with 2 pins. If you want a speedo, you either need to connect it to a timer for input capture or use external interrupts, you only need a single pin for this.
As TLHarrell mentions, for brakes, the best trigger is a micro switch, although you'd want them in parallel not series.
For the lights themselves, if you want to adjust brightness levels, you need PWM pins, there should be enough PWM pins for you to use for all 3 conditions (brakes, and turn signals). Just remember you need to program the arduino without delays.
Lastly, there is a trick you can use to use a single pin for the turn signal buttons, but it's not recommended and you most likely have enough pins, no sweat. You can set up a resistor divider and use the switches to 'short out' resistors and by measuring the voltage over a resistor, you can work out which switch is being pressed. However it's prone to noise, which is a factor.