Everyone has their style of shooting.
I personally don't find Program (P) mode very useful, but the name itself is pretty much self explanatory. It's basically Auto+ for me. Nonetheless it has its uses for sure..
Keeping in mind the 3 parameters in photography:
1) ISO - this affects amount of noise and also the speed of your sensor in capturing light.
2) Aperture - this affects the depth of field (i.e. how much of the picture is in focus) as well as how much light your lens can capture during a given set of time.
3) Shutter speed - this affects the time the shutter is open for and the amount of motion in your picture.
The end goal in a photograph, when it comes to technicalities, is exposure. There is no right exposure but to get a certain exposure you will have many sets of ISO/aperture/shutter speed combinations. They all exert a relationship on each other. For example when you need to reduce the shutter speed while retaining the depth of field in a picture (perhaps because you can't get a good handheld sharp photo with existing shutter speed), you then keep aperture, reduce shutter speed and dial up ISO. So on and so forth. There's a wealth of information on this, go read up if you don't already know. Now with THAT in mind...
M/Manual mode offers the most control as you can exercise discretion over every single setting. Useful when you need to fix the exposure, for example, if you're taking a set of images to stitch iThe downside is that you have to fiddle with more things. Not very useful for situations where you need to change the settings fast with less thinking, e.g. when shooting in constantly varying lighting conditions. In general, good for landscape photography.
A/Aperture priority mode prioritizes the aperture. So your concern here is the amount of depth of field. The camera then selects the aperture speed for you, and you may or may not fill in the blanks for ISO speed... Useful for situations where your priority is shallow DOF for sure, e.g. portrait photography. Downside, less control. You're releasing more and more control to your camera meter, which can be fooled at times.
S/Shutter speed priority prioritizes shutter speed. So your concern here is the amount of time the shutter is open for. The camera selects the aperture for you, and you may or may not fill in the blanks for ISO speed... Useful for situations where you need to capture something FAST, or if you don't think you can handhold above certain shutter speeds, e.g. sports photography. Downside is also less control.
The above are just generalities and everyone's style is different. I know wedding photographers who prefer to use manual mode all the time despite the fact that they're in a fast paced environment with very tricky lighting situations, because these situations fool the camera meter too! I also know landscape photographers who use aperture priority because well, they just prefer it.
What should you use? Just try la. Then you can answer the question yourself.