This is the time to visit botanical gardens (there are two in Kyiv — a small one next to Universitetska metro station and a large one near Arsenalna station), begin sunbathing, travel to Crimea (not in the summer when it's often intolerably hot), and take road trips to all those obscure destinations you've always wanted to see around Ukraine.
One of the few things you'll need to worry about during this blissful but brief period is ticks ("клещи"). May seems to be their busiest month, and they tend to taper off through the summer. Ticks around Ukraine have been known to carry encephalitis, though cases seem to be very rare.
After spending time outside (on the grass, at botanical gardens, in the forest, etc.), you should check your body over for ticks. They can be a bit hard to see, and prefer areas where the skin is soft — behind the knees, around the tops of thighs and in the groin area, around the armpits, etc. It usually takes them quite a while to crawl up to a good spot, so you often have a good half hour or so to nab them.
Ticks can be removed with tweezers (taking care to grab it as close to the skin as possible), by applying oil and gently rolling the tick over and over with a circular motion (may take 5 seconds to 5 minutes), or by using special "tick tweezers" which one can find in the U.K. and some other places.