Hi,
with just three days I'd only stay at one place .
Land at the airport and head for kandy. Nearest to the airport to travel up there is a taxi, followed by a bus, followed by the train.
If you go bus from the airport, there's a free shuttle bus goes from just outside the airport front door, to Katunayake. ( about 10 minutes ride ) from Katunayake you can get a bus for Kandy. About 3 hour ride.
Look at the last post for a couple of places to stay.
From Kandy, a day trip to Sigirya and the cave temples at Dambulla.
a day trip south into the hill country. You can do that on a bus from kandy to Newara Eliya stopping at the Ramboda Falls and a tea plantation for a tour..... Stroll around and look at newara Eliya and then get the train back to kandy. This trip is really to see the hill country scenery.
last day, Botanical Gardens, Temple of the Tooth and Kandy city streets.
Then, On your way back to the airport you might just find time to stop at Pinawella Elaphant Orphanage en route.
And, yes,,, it's safe.. Even during the height of the war, as far as tourists were concerned the buses were more dangerous than the Tigers..
Rod.