As others have said, there is a fine art to TIA design but it doesn't have to be that hard.
The main limiting factors will be the photo diode capacitance, op amp capacitance and how much gain you want as the gain resistor determines the pole frequency in combination with the capacitance.
Then there's noise and stability to consider but both can be calculated here:
http://www.jensign.com/transimpedanceamp/About op amps, OPA355 family work well with a supply of 5V or less but are a bit noisy and probably will be only marginally better than what you are using now.
With split power supplies, it's hard to beat OPA659 in a TIA unless you do some trickery using a cascode at the diode junction to eliminate the effect of the capacitance.
For high speed, always apply as much negative bias to the diode to eliminate the capacitance.
And you must use a low input bias op amp FET or CMOS, not bipolar input.
What diode are you using and in what configuration?
What supply do you have to play with?
Is noise important?
Does it need to be linear or is this for a data link or similar?
How much TIA gain do you need (what is the value of the feedback resistor)?
EDIT: Alex Nikitin previously suggested an LTC6268 and this looks like a really low capacitance op amp (the lowest so far), worth considering.
And OPA657 is 1.6Ghz bandwidth but may not work as well due to higher capacitance and it's also not unity gain stable which might prevent it's use depending on your configuration.