There are no bus station at International airport Simferopol. So if you plan to use the public transport to go to Yalta, Sevastopol, Feodosia etc. you first should to go to the bus station. Note, airport situated at the North end of Simferopol, in about 10-12 km. from city center (see here: map of Simferopol). There are several bus stations in Simferopol. The most convenient bus station is situated at the Simferopol train station. From here buses go to Yalta, Sevastopol, Yevpatoria, Sudak, Kerch etc. every 10-25 minutes.
Public transport stop is situated just in 50-60m. from the main arriving terminal. Take any bus or trolleybus (electric bus) and it will definitely go through the train station.
Another option to go from airport to one of cities in Crimea is to take a taxi. Normal price for standard car is something around 12-15 times more expensive, than a price of public transport. At airport you will be «attacked» by a huge number of taxi drivers, offering their services. I would say, booking a transfer in advance is always better and cheaper.
Bus station is situated just opposite the train station. It's easy to find it (you will definitely see it on your way from railway) and to use it. In high season there are buses already at something around 5:30 am. But even in high season, there can be problems of using public transport later, than 20:00.
On the photo railway is situated just behind arcs and bus station before arcs and big tower with clocks.
Main bus station is situated on the Northern end of city. Any trolleybus (electric bus) will finally take you to the Yalta's bus station. From here there are plenty of buses to Simferopol and Sevastopol, but only a few buses to Bakhchisarai, Sudak, Feodosiya (it can be only 1 or 2 buses a day!!!).
Another small small bus stop is situated in the city center (when road from main bus station - street Kievskaya - see map of Yalta - reach the Black sea and turns to the right, this bus stop will be on your left.) From here you can take a bus to Vorontsov Palace, Cable car lower station, Nikitsky Botanical Garden and some other destinations.
Main bus station is situated just opposite the train station in Sevastopol. From here you can take a bus to any of big cities of Crimea. If you plan to visit Balaklava, so you first should to go (by any city public transport) to the stop «5-th kilometer» and from here take a bus to Balaklava.