Well, you hadn't told us that yet, so how were we to know?The thing is the point of the project is to have the motor run on a recharable battery. Connecting a 9v or an external source similar to it might be counter-intuitive to what I am trying to accomplish you know? I appreciate the idea though thank you!
Anyway, when testing things, it's generally a good idea to work in small steps and test incrementally. In this particular case, you don't really know how well your Arduino/relay combination is working yet, you don't know if your rechargable battery setup is working yet, and you don't know if your new motor will work well on your rechargable battery.
Instead of mashing a whole bunch of unknown things together at once, why not change just one variable at a time and observe the results? That way, if it works you can proceed to the next step. If it doesn't work, you know the problem is in the one variable you changed. Otherwise, if you change everything at once and it doesn't work, you have no idea which change is the problem.
I think doing a quick test with the 9V battery is smart. That will prove that you have the Arduino relay control working properly, and that you understand which I/O terminals on the relay board you need to connect in order to have the relay successfully control the motor. Once you've established those things, add the new variable of trying to run this whole thing on the rechargeable battery system.
Of course, you're free to skip as many intermediate steps as you want, and if it all works perfectly then you'll have saved a little time by skipping ahead... but if it doesn't work, you'll have no idea why, and the first thing we'll suggest is backtracking and doing incremental testing.