How do I turn on a device with one momentary switch and off with another
You need a latching relay

This type of relay does pretty much exactly what you want, changes between two outputs states as a result of a short pulse on it's input. If you get a "dual coil" or "double winding" type then there will two inputs, one called "set" and one called "reset".  The set input turns the output on, and it stays on until the reset input is used, which turns the output off.  

There are some designs of latching relay which latch mechanically (as opposed to magnetically), and these don't need any power to stay on, so wouldn't drain the battery when the bike is off.

The relay will need it's set and reset inputs to be on for a minimum period in order to switch, probably some milliseconds. You can either include some digital circuitry to make sure that is the case, or just hold the buttons for half a second when you press them.