Connecting a photodiode in reverse bias to an oscilloscope
Here is the recommended circuit from the datasheet:

Put your probe on the plus, and the ground on the -.  The noise filter won't hurt but is probably unnecessary with a battery as the power source.
And here is the formula for calculating RL:

The diode itself acts as a capacitor with 24 pF capacitance.  When you hit it with a photon, you get one electron of photocurrent, which flows through the load resistance forming a low pass filter with time constant R*24pF.  You need the resistor to convert the current into a voltage.  A larger resistor would give you a larger voltage, but you need the resistor to be small otherwise the rise time will be long.
If you plug in the recommended Thorlabs values, you will get that the rise time is actually 2 ns, not 10ns.  I have used this photodiode and it is in fact faster than the datasheet implies at 50 ohms, at least in all the units I got from them (perhaps they have different suppliers or simply didn't measure with a fast enoughs cope).  You could take a chance and use a 100 ohm resistor which will give you 2x the signal amplitude, or play it safe and use the 50 ohm resistance they recommend.
As requested, here are the calculations for that diode:

Plugging in the values of 50 ohm and 24 pF gives a Fbw of 132 MHz and a rise time of 2.64 ns, or about 3.8 times faster than speced.  You could therefore use a 180 ohm resistor and get 3.6 times the amplitude (although the 1 nA of dark current will also lead to 3.6 times larger amplitude). This calculation doesn't depend on the intensity of the light source, but for a fast photodiode a relatively bright light source will be required for that bandwidth to have acceptable SNR.  For low light measurements you would use a source such as a PMT or APD that has intrinsic gain (more than 1 electron per photon).