Turkey has a great, simple public transport system: the Dolmus. They are minibuses, driving along fixed lines (but often willing to make a little detour for you) between small busstations. There is no timetable, but they go very frequently, often every 5 minutes. They stop wherever people want.
Karga Yuvasi is within 100 meters of 2 city lines (number 1 and 2). The dolmus station of Kuşadası is at a 10 minutes walk. From here dolmuses go to all directions. Destinations far away may require a transit at another station: the farther the destination, the bigger the bus. Turkish people are generally very helpful to help you find the dolmus you need. Travelling by dolmus is very authentic and cheap.
Taxis are everywhere you look. It's best to negotiate a price (or an indication) before getting in. All drivers speak English well enough.
Many companies offer cars and motor cycles for rent. International European and British drivers licenses are valid in Turkey.
There are not a lot of trains in Turkey and you can embark only a few time a day. The closest train stations are at Söke and Selçuk. Travelling by train however, is quite adventurous and costs almost nothing...