The Bakhchysaray region has several cave towns and monasteries. It is considered as one of the more important ones. Since I only visited this one I can't go and compare. But the scenery is fantastic and the way the architecture is combined with the rocks is interesting.
Since it is an active monastery you should wear appropiate clothing. Prepare for some hiking and uneven paths and in the summer bring plenty of water. At the entrance there are a few restaurants with some good Tartar food. There are many little stands everywhere trying to sell you local products. The most famous are the perfums, they sell them everywhere in the Crimea but here you get the best quality for the cheapest price.
Getting here on your own might not be that easy, I went with a group of people and we hired a mini-bus.

A nice monastery in the mountains. Be careful with taking pictures of icons, people don't like that. If you have time, it's a nice walk towards Chufut Kale. It's also worth it to walk down the mountain towards the Khan-castle downhill in Bakhchysaray.