This is all pretty straightforward.
First, signal to noise. Get an RMS AC meter with a bandwidth of about 10 MHz, and measure the TIA output when the detector is dark. (Make sure to account for the meter's equivalent noise. AC meters are notorious for picking up crap in the ether.) This will give you your noise spec. Then signal to noise ratio will obviously depend on your signal.
Next, get an oscilloscope, and build the following circuit:
simulate this circuit – Schematic created using CircuitLab
Mount the LED and PD as close as possible, with a light shield around them.
Now drive the LED from a function generator, and with a pulse or square wave (watch out for reverse biasing the LED) you can determine the rise time of the LED and PD together, which should be less than 100 nsec. Make sure that the PD is not being overdriven if your LED is very bright. Look for maximum current on the data sheet (10 mA is a good first assumption), then make sure that the output to the scope does not exceed the iR product of the max current and the load resistor.
Now connect the LED to your PD/TIA and measure the output. You should see the rise time is much slower.