2 to 3 days in Romania

I have been to Bucharest 4 times. I also visited some other cities during business trips to this country.
When I go to Romania in April, I want to stay 3 additional days. My flight goes from OTP, but other airport is possible.
Is there anybody whis some suggestion what is worth to visit outside the capital (I prefer public transport, but if necessary I can rent a car).

The area most popular with tourists (far more popular than Bucharest) is southern Transylvania (the region around Brasov and Sibiu).

You could head to Brasov (fly to Bucharest then catch a train) and use it as a base for your 2-3 days. You can visit medieval quarters, hike into the mountains right next to the city or try to see a few of the nearby sites: Peles palace in Sinaia (about one hour away by train), Sighisoara (two hours away), Bran castle (~40 minutes), Harman or Prejmer fortress churches (~30 minutes) etc.

A day-trip from Bucharest to Sinaia
http://www.romaniatourism.com/bucharest.html#daytrips
is easily doable by train
http://www.romaniatourism.com/transportation.html

You can also include Brasov in a long 12 to 14 hours excursion.
In case that you can spend a night in the countryside,
hotels are less expensive in Brasov or Sinaia.

Ideas for other Romania itineraries:
http://www.romaniatourism.com/sample-itineraries.html

As you have just 2 to 3 days it is difficult but maybe for another time I woud recommend you the Maramures region. This rural area in the northwest of the country is very special, the wooden churches and houses, friendliness of people, nature etc. There traditions are still alive. You can fly to Satu Mare or Cluj Napoca and rent a car.

I spent 4 nights in Sibiu and was utterly charmed by the city and its surroundings. We rented a car for two separate day trips, one to the Fagaras Mountains for a hike and one to the Hunedoara area to see the Corvin Castle and Roman ruins to the southwest. April may not be the season for a mountain hike, but Sighisoara is a doable day trip. I haven't been to Brasov or most of the rest of Romania (yet!), but there are flights to Sibiu, and the train from Bucharest is about 5 hours if I remember correctly.

If you want a 2 hour train ride out of Bucharest, and don't have your own transport once in a place, then I sugest Sinaia, a ski resort town north of Bucharest. Not very sure about the weather in April, but enough accomm in the town, Peles castle is a MUST-SEE, gondola rides to the top of the mountain (Corta 2000), and/or walks on/up/down the mountain from the town. But I also go with Sibiu and Brasov (B is further up the line from Sinaia, and a little bigger). The Transfagarasan Hwy (if you rent a car) is stunning. Easily done as a day trip out of Sibiu. And Sibiu is also about 2 1/2 hrs from Bucharest, but via Brasov and Sinaia.

Sorry, correction: Sibiu is more than 2 1/2 hrs out of Bucharest, as you have to go via Sinaia and Brasov. I did this route last year June. LOVED Romania.