To Khan's Palace/Chufut Kale from the bus station
From the bus station we got directly on a #2 marshrutky (1.5 uah) to get to the Khan's Palace. It was fairly evident when we saw the Khan's Palace with it's minarets that we needed to get off but if you ask the driver he will tell you when you are there.
After we were done visiting Khan's Palace, we got on another #2 marshrutky going in the same direction that took us to the entrance to Chufut Kale (cave city), the van turns around when it gets as close to the caves as it's going to get. You still have quite a bit of a walk to get to either the Upensky Monastery or Chufut Kale, you will come across the monastery first and then you have a further hike to get to the entrance to Chufut Kale. The walkway is paved until the monastery and for a stretch on the way to Chufut Kale but the last section is paved with uneven rocks. Although the young Ukrainian girls seemed to have little trouble managing it with heels on, you might want to wear shoes more suitable for uneven surfaces.
By bus from Balaklava
Our location in Balaklava added a bit of time to the journey, we had to take marshrutka #9 to 5km (2 uah), then another bus or marshrutka #17 or #20 (1.5 uah) to the Sevastopol bus station and then another marshrutka to Bakhchysary (12.02 uah). You buy your tickets for the marshrutka from the driver, most of the time paying when you exit the van. For buses you need to buy tickets from either the driver or the ticket taker, sometimes they rip them, sometimes you have to punch them. We purchased the ticket from Sevastopol to Bakhchysary inside the bus station, that way you can make sure you get on the right bus and the one that is leaving the soonest. The ticket has the time it leaves and the clerk should tell you which number stop you will leave from. Plus you can confirm in the van's front window that the van is going to Bakchysaray as long as you know the cyrillic spelling. If it's direct service it may make stops along the way at bus stops or if someone has their hand out but everyone left will get off at Bakchysaray. The journey from Sevastopol to Bakhchysaray took about an hour.
Bus from Simferopol
Busses are going to Bakchysaray every 10 minutes from Simferopol. You can get there by bus from the trainstation of Simferopol, but better is to take a bus from the busstation. It will take 30 minutes and costs 6 UAH.