Phin and I have had strong recommendations from several people to see Kiev Pechersk Lavra (Києво-Печерська лавра), a large complex built up around a cave monastery that was founded in 1051. It was a beautiful and awesome place that we visited twice.
The caves are open from 8:30 till 16:30 daily, and we were too late the first time we went, so we went again this morning. We were able to attend the last few minutes of a wedding, hear monks singing on several occasions, tour the cave monastery and several churches, visit a few exhibitions, and see the "microminiatures" exhibit that is a little out of place but really amazing.
Here is some of what we saw outdoors:
A very beautiful place.