Hi Jordan, I live here in Ukraine and the best way of going around I would say it's by train, the prices are fair enough and they are not bad, rather say even comfortable, with exception of toilets. The price will be around 14EU for 2nd class, cabins for 4 people and door or around 9EU for open cabin, 6 people sleeping - be prepared to interact with people and vodka...
You can catch flights, with Aerosvit, ukrainian airlines that sometimes make promotional prices.
Must see in Crimea I would say that you need some time over there, Sevastopol and the nuclear submarine base as well as the landscape are nice, just beaten by Sudak, check images over internet, is a really beautiful rest place with old ruins, canyons, nice beaches, calm with tourists but not as packed as Yalta, Alushta or others. But Yalta has a really nice landscape and hiking as well.
Staying in Ukraine is the tricky thing, plan a little in advance, check hostels and in Crimea hostels open only around may so you can basically book only around may, other options might be over priced with bad service/information/quality. Check tiu hostels, a bunch of foreigners running fun places in different cities, i staid with them and worth, even more near train station, amazing cheap places to eat and hang out, with the same walking distance from any sight seeing.