Transimpedance Amplifier Photodiode
The datasheet of the photodiode contains many useful bits of information such as:


Max thermal dissipation (limits the power incident on the photodiode)
Dark current (this is the signal that is present in the absence of light)
Junction capacitance variation with reverse voltage
Responsivity (A/W) at various wavelengths.
Breakdown voltage


The datasheet of the op-amp also contains useful info such as:


Gain-Bandwidth product
Minimum stable gain (in voltage to voltage mode)
Bias current (some high-GBWP op-amps draw significant current at their inputs)


A good way to model photodiodes is as a current source (whose current depends on the incident light x responsivity) in parallel with a resistance (which represents the dark current) in parallel with a capacitance (the junction capacitance). The junction capacitance decreases with increasing reverse bias.

A lot of these things can be learned by looking at Application Notes (even if you have little background in electrical engineering, although you should know how op-amps work). Here's a list of my favorite resources:


http://www.osioptoelectronics.com/technology-corner/application-notes.aspx
http://ecee.colorado.edu/~ecen4827/hw/hw1/PhotodiodeAmplifers.pdf


If all you need is low noise, things are easy. Reduce the bandwidth by hanging a huge cap either across the feedback or at the output, and the noise will be averaged out. Also remember that if you are detecting a laser beam, there will also be shot noise, so you may want to detect as much power as you can and use a large photodiode to dissipate all that heat.

If you need a lot of bandwidth, things are a bit more tricky. The bandwidth depends on the junction capacitance, on the resistance and the GBWP of the op-amp. The junction capacitance depends on the type of photodiode (Silicon, InGaAs, GaP...), the size of the chip (more area means more capacitance) and the reverse voltage. So you would want the smallest photodiode you can deal with without damaging it due to the poor heat dissipation. Or maybe you can think of putting a Peltier cooler or something.

The National Semiconductor presentation above discusses various ways of trying to keep the bandwidth high. There's also some info on Thorlabs' website.