Diodes are current-based devices, so you need a circuit to convert the small photodiode current into a voltage.
Check out this presentation for a simple op-amp circuit implementation, slide 9.
ecee.colorado.edu/~ecen4827/hw/hw1/PhotodiodeAmplifers.pdf
As for calculating irradiance, I found some useful equations on this site: **broken link removed**
Eqn 4 states: Irms = R*Ee*Ad
Where Irms is the average current out of the diode,
R is the Responsivity of the diode in Amps/Watt,
Ee is the irradiance in Watts/cm^2,
and Ad is the illuminated capture area of the diode.
The op-amp will give you a conversion from current to voltage. If you measure that voltage and scale it according to the op-amp's conversion gain, with a little algebra (and the photodiode characteristics from the spec sheet), you can calculate Ee (irradiance in W/cm^2).
Be sure to watch out for the units and make sure they cancel out correctly, so you know if you are in W/cm^2 or W/m^2 (depends on what's given on your diode's spec sheet).