The are a few ways to get to Donetsk. You can try to fly in directly (the airport code is DOK) usually through Kiev, Munich, Istanbul or Vienna. There are also options if you're in Ukraine - by train, air or by car. The drive from Kiev can be anywhere from 6 to 8 hours depending on how fast you drive, how many stops you make, etc.
Public transportation is cheap, but can also be crowded. Be prepared to stand up most of the time, and you may have to make waves when you want to get out. There are three main types: the trolleybus, tram and minibuses. Both the tram and the trolley bus will get you just about anywhere in the city for 50 kopek (around 5 Euro cents). This price is for as long as you want to be on the tram/bus. At the central market there is one of the many depots. These forms of transport stop at every stop on the route.
The minibuses can be found throughout the city. They are like the buses, but faster,smaller, and they don't stop at every stop. If you are on board you must yell to the driver that you want out at the next stop, and if you are waiting for one you must flag it down. Depending on the driver, these can get very crowded. You will pay about 2 Hrivnya.
Most tourists find the taxi to be their best option. Taxi cabs are on every corner and they are relatively cheap. To go across the city, you can expect to pay 30 Hrivnya (about 6 dollars), but the average amount you would pay to get around the center of the city is 1-2 dollars.
There are numerous dispatch taxi services. For example, 381-0000 or 0-67. Keep your eyes open for more numbers, usually printed on the side of the cab in large font.
Also, on the first ride in the taxi, ask the driver if he would like to drive you around on a regular basis. Usually he would be delighted to do so and would give you his ID# for dispatch (pozivnoi) or his cell phone so you can call him directly. This way you can have a chauffeur for the time of your stay.