10 amps, ok. Amperes are not per-unit time though (well technically you can define them by coulombs per second, but preferred is force between two parallel conductors.. nevermind!) You are maybe talking about amp-hours? battery capacity. Theoretically a 24V 10Ah battery would let you run your bike for 1 hour. But this is a bad theory, in practice the battery wont last that long. Looks like you understand that... 2 parts to this. You need a throttle/PWM circuit, and a motor controller circuit. If it was me I would use a microcontroller with an ADC to read a potentiometer and output the PWM. Then you have the ability to program in other fun features. But you could also use something like a 555 timer IC I bet.
As for the motor controller, I have seen combat robotics motor controller plans that would handle these specs. Try searching the web some more. I dont think you need a full H-Bridge though right? The bicycle only needs to go forward and the motor only needs to spin in one direction? That simplifies your motor controller greatly. Use a big N channel power MOSFET as a low side driver.
Batteries in parallel allows you to run longer (or draw more current). No modifications should be necessary, your circuit will still draw the same amount of current. You might want a fuse.
good luck