While in many cases, painting a ceiling is a daunting task in and of itself, having to edge around the recessed lighting makes the process that much more difficult and won't give the smooth, professional look that you're hoping for. Luckily, removing the recessed light trim is a simple process. The recessed light trim is simply held in place by springs in the frame of the light.
Turn off the light you will be working on.
Use your flathead screwdriver to get under the edge and give a firm pull. The recessed light trim is simply held in place by springs in the frame of the light. Unless the trim has been glued in place (which is not typical), the trim will come down.
Once you have completed painting the ceiling (and the paint has dried), just pop the trim back into place by lining it up and firmly pushing it back into the slot. It should snap back into place fairly easily.