First a schematic or other diagram of how everything is wired would be very helpful in most cases when you are asking for advice on this forum.
IANfinity:
So we bought a small DC motor (not sure of the specs)
If it has any markings, look them up on-line. You might be able to find a data sheet for it. After all this is the Information Age... 
IANfinity:
I have a 9V battery which is to power an Arduino Uno AND a DC motor.
Regardless of your other problems if it's the standard rectangular 9V battery, you want a different power supply. A 9V battery usually has less than an 700 mAh,(or it can also be expressed as 0.7 Ah), of charge capacity. The units "mAh" or "Ah" stand for "milliAmpere-hours" or "Ampere-hours", in other words how much current the battery can supply for one full hour. This is enough to power an Arduino and a few other things like LEDs for several hours, but it wouldn't power most DC motors for long and at least some it couldn't supply enough current to get them moving (though it might not be the case with your motor).
On the other hand, common alkaline AA usually have a capacity of over 2200 mAh. They have a nominal voltage of 1.5 VDC (nominal because the exact voltage of any battery varies a bit based on how much charge is remaining), but six of them can be put in series to get a nominal 9 VDC. By the way, all the standard "letter" sized alkaline batteries have nominal voltages of 1.5 VDC. The only difference is the maximum charge capacity and as you might guess the capacity increases with the battery size.
IANfinity:
What kind of DC motor do I need?
The primary requirement will be "Can the motor do what I want it to do?", in this case spin your weight reliably with enough force to tip over the can. Any other requirements will be imposed by limitations like volume, weight, and power constraints (which in this case shouldn't be a problem). With a few extra components and the right programming an Arduino can control almost any DC motor.
IANfinity:
What kind of transistor do I need? What kind of resistors do I need, if any?
The transitor will have to be able to handle the voltage and the current levels necessary to power your DC motor, while still being able to be controlled by the output of an Arduino pin, which can only output 0 or 5 VDC @ a maximum current of 40 mA. The number and rating of resistors will be determined by what's necessary to properly bias the transistor.
You should also have a couple of diodes as well.
Here's a link to a tutorial about working with small DC motors and Arduinos. It's written by an active forum member that goes by "Grumpy_Mike".