The first thing you need to do is buy a decent saw blade. Buy one that's specifically designed for cutting laminates and plywood. Plan to spend at least $25-30, if not more.
Next, build yourself a sawboard. There's a sticky post in this forum that describes how to do it. Not only does it serve as a guide, it will also prevent the material that is directly underneath the sawboard from splintering or chipping.
When you're ready, make all of your cuts on a sacrificial surface. You want the entire cut to be supported by this surface: it's what prevents splintering on the back of the cut. If you have any old plywood, MDF, or melamine use it. You can also use large sheets of hard foam insulation. Don't try to cut through this backer! Set the depth of the blade so that you're only cutting slightly into the backer board. That way you'll be able to make many cuts without the backer board falling apart.
Finally, make some test cuts and then plan your cutting sequence based on the results. If you do everything I've suggested, you should get pretty good results, but there may be some chipping on the top of the piece on the part that the sawboard doesn't cover. Make your cuts so that these chips won't be exposed (e.g. they're on the inside of the cabinet). You can also try using masking tape across the cut line. though it's a bit hit-or-miss in my experience.
Finally, pay attention to the quality of your cuts as you're moving through the project. MDF and melamine dull saw blades quickly. In addition, you may find that little bits of melamine adhere to the sides of the saw's teeth. If either of these things happen, you'll suddenly start to get lots of chipping. Examine the saw blade and clean or resharpen it as needed.