Sinaia can easily be done as a day trip from Brasov, it's just one hour away by train and there are more than 20 direct trains daily (practically one per hour).
Doing Sibiu as a day trip is a bit more complicated, because it's further away from Brasov (about 2 1/2hours with a fast train) and most importantly because there aren't many convenient ways to get there early in the morning (there is a train leaving at 6AM from Brasov and arriving around 10AM to Sibiu, but it's a "Personal" type train, which should be avoided).
However, Sibiu is absolutely worth visiting and you should probably spend more time there, it is one of the most important historical cities in southern Transylvania, it has a beautifully restored medieval downtown area, plenty of museums and various other tourist sites.
To compare, Sinaia is a small mountain resort, the main tourist sites there being Peles castle and the monastery; nonetheless it does offer many opportunities for mountain-related activities.
So if you have to choose one of them and you're not particularly interested in hiking/climbing , I'd say choose Sibiu.
For the flying out part, if you leave from Brasov, it's much easier to get to Bucharest (3 hours away) than to Cluj (5 hours away); if you spend a few days after Brasov in Sibiu (which I suggest once again) and leave from there it's easier to get to Cluj than to Bucharest.
Don't worry about Bucharest, there are plenty of things to see/do here and it's a reasonably safe place; but Cluj is a very interesting city too.
