To shop you will most certainly need local currency hryvnia. British pound, Euro, US dollar and other currency exchange points are very common in cities, and the exchange rate is usually very fair except in Kiev, where the exchange rate is higher compared to other cities. However, sometimes and in some banks there are problems with cash deposits or that is the official version, so do not exchange too many dollars unless you're traveling to the more provincial areas. When doing person-to-person payments you might be able to pay in Euros or US dollars, as those are widely recognized, and you might in fact get better rates than in official exchange points. However, be careful, because it's not legal to make payments with foreign currency.
If you want to buy any kind of artwork paintings, Easter eggs in Kiev, the place to visit is Andriivskij Uzviz ÐÐ½Ð´ÑÑÑÐ²ÑÑÐºÐ¸Ð¹ ÑÐ·Ð²ÑÐ· in Ukrainian, ÐÐ½Ð´ÑÐµÐµÐ²ÑÐºÐ¸Ð¹ ÑÐ¿ÑÑÐº in Russian.
BE AWARE THAT IT IS ILLEGAL TO TAKE OUT OF THE COUNTRY ANY ITEMS OF HISTORICAL IMPORTANCE including badges, medals, icons, historical paintings, etc. This law is strictly enforced at all exit points of the country and one risks heavy fines and/or imprisonment.
If you don't want to have trouble exchanging money have at yourself one of these currencies: EUR, RUB, USD. In South and East Ukraine, it is almost impossible to exchange other currency.