edit: Whilst I have one recommendation for the op amp I'd really like to know how to pick them myself, what sort of specs should I be using to search for them?
Some pertinent op amp characteristics:
Gain-bandwith product (GBW) - This determines the gain and bandwidth the circuit can have. If you divide GBW by the circuit gain, you get the maximum bandwidth the circuit can have.
Input bias current - This determines the amount of output DC offset you will get with the input and feedback resistors you use. You multiply the input bias current by the input resistor and the circuit gain to get the output offset voltage with zero input.
Input offset voltage - You multiply this offset by the DC circuit gain to determine the amount of DC output voltage you will have with zero input. You add this to the offset from the input bias current to get the worst-case total output offset.
Operating voltage - Determines the power supply voltage(s) you need.
Operating current - Determines how much power the device uses.
Slew Rate - This is how fast the op amp can swing the output voltage. It often limits the voltage swing at higher frequencies. For example the slew rate of a sine-wave is 2Pi x Vp x f where Vp is the peak voltage and f is the frequency of the sine wave. When this value equals the op amp slew rate you can go no higher in frequency or amplitude without distortion at the output.
Input/Output Type - Rail-to-rail can have inputs and outputs at the supply rails. They are generally what you want if you only have one supply voltage. Non rail-to-rail usually can only operate within 2 or 3 volts of the supply rails. They are more typically used with dual power supplies.
There can be other characteristics the may be important in some applications, such as the input noise, but the above covers the basics. Often you have to do a trade-off between characteristics for a particular application. That's why there's no "one size fits all" in op amps.