Tip #1 - One thing you can do is to pull the trim straight over the finish nails. Trying to remove finish nails is frustrating; will gouge the trim; and will take forever. Instead, gently pry the trim up and let the finish nails stay in place. Then, knock those suckers back into the walls. Or, if you're a stickler about such matters, you can remove those nails with a lineman's pliers.
Tip #2 - If it's an older house, you can count on your walls and trim having multiple layers of paint. When you rip off trimwork, you will also rip off significant amounts of the wall itself. Remedy this problem by scoring the intersection between trim and wall with a utility knife.
Tip #3 - Count on wastage. No matter how careful you are, you will still end up breaking and splintering your trim. You can't save all of it. If you've already decided to install new trim rather than saving the old stuff, then obviously your work will go much faster. Crown molding is exceedingly difficult to remove without breaking.
Tip #4 - Safety! When removing trim, you'll often end up with protruding nail (even if you've already decided to pull the trim over the nail, as detailed above). These protruding nails are just begging for a nice, soft foot to stab into. I know: I've been there before, and it's an experience that still makes me cringe to this day. Immediately dispose of those boards in your dumpster. Also, be sure to wear work boots.