Measurement of output current of a photodiode
I'm not sure weather to use just an oscilloscope, a microcontroller (Arduino?) or something else to get the data and get them plotted.

Different measurements need different instruments.
Rise and fall times: you need an oscilloscope
I-V curve: Ideally you want an SMU (source-measure unit) such as the Keithley &quot;Sourcemeter&quot; products.
Responsivity: A good voltmeter
Signal-to-noise ratio is going to be your trickiest measurement. It will be challenging to measure the SNR of the actual photodiode rather than of your transimpedance amplifier and other receiving components. If the SNR is very bad, an oscilloscope will be adequate to measure it. If the SNR is very good you will need some careful design and possibly some expensive equipment.