Not sure why you want to use visible red light. Is that just because it is what you have, or is it part of the demo?
For the basic design steps, you have to consider:
A. You need to deal with ambient light vs. laser intensity.
B. You need to deal with interfacing between the photodiode and whatever you want it to "trip".
C. You may want to consider advanced techniques to improve #1.
Assuming the laser is always on (#C is discussed later), you probably need to have a threshold calibration to set a threshold for the laser light level.
The laser needs to be bright enough to make a significant difference to the ambient light. This is essentially your
signal to noise level.
A. There are several simple optical methods to help you out with ambient to laser ratio:
1. Get a filter for the laser frequency to put in front of the photo diode. <---- Probably Required
2. Put a neutral density filter in front of the laser to avoid saturating (overloading) your detector. <--- Kludge
3. Add a lens system that focuses a narrow area on the detector. <--- Good
4. Something as simple as a cardboard tube can reduce adjacent light sources <--- quick and simple
B. There are many different circuits ranging from very basic and insensitive, to very sensitive. The link from davenn shows they range from simple transistor based circuits to opamp based circuits. The most basic opamp principle for a photodiode is shown below. Measuring the voltage output from such a circuit would allow you to experiment with The ideas in A and give you some idea of what the final design needs. The value of Rf depends on the diode and light levels.
Essentially when light falls on a reverse biased photodiode, the leakage current increases and the opamp output voltage changes to supply the current.
This circuit is an over simplification to show the principle.
You may want to search for a module. Such as
http://www.lightinthebox.com/lm393-light-sensor-photosensitive-sensitivity-light-sensor-module-for-arduino-free-dupont-cables_p1023968.html <--- looks adequate!
https://www.amazon.com/dp/B00JRNN1FW/?tag=pfamazon01-20
Although, I didn't see schematics for either of those.
C. The next level of sophistication involves chopping or modulating the light at a particular frequency and building your detector to filter for that frequency. I think you need to understand all of the above and determine a true need before you head down the modulation path.