Hi, a happy 2014 to you as well. When finishing a basement, the layout of existing mechanicals will most often determine where you can use drywall or a drop ceiling. There are certain electrical, plumbing and HVAC components that by code cannot be permanently covered up. A drop ceiling allows service to these systems to be performed if necessary. Drywall (though a little more costly) is often more preferred aesthetically. As far as the transition between the two, in general you will need something to fasten the drop ceiling track to. If you look closely at the transitions in these photos, you will see that there is a drywall return about a 1/2" lower than the drop ceiling in order to accomplish this.

Did you decide to combine drywall and the drop ceiling? If so I would like to know how it turned out. We are trying to decide the same thing currently.

There are some great ceiling tiles (even over your existing) you can install that would match your decor and still give you access to the plumbing. See http://www.proceilingtiles.com/Stratford-Ceiling-Tile-White.html