Virtually everything you hear on the street in Dnipropetrovsk will be in Russian, not Ukrainian. You can probably get a dozen Russian TV channels, none of which are in Ukrainian.
As you noted, Ukrainian uses the letters І and Ї that are absent from Russian. The Ukrainian Є is also missing from Russian, while the Russian Ы is not used in Ukrainian.
In terms of the spoken language, its more complicated because the Russian spoken in Ukraine is a southern dialect with transitional features that make it closer to Ukrainian, such as Г pronounced as an English H (not G), and full pronunciation of O even when not stressed. Furthermore, many Ukrainian speakers intersperse their conversion with Russian words such as Хорошо and Молодец. Some younger people code switch, starting a sentence in one language and switching to another.
One think to look out for is Russian words containing “O” that often transform to “I” in Ukrainian, for example Комната becomes Кімната. The Ukrainian word “Добре”, meaning fine, is not used in the same way in Russian (Добрий человек is a good man, but the Russian equivalent to Добре is the ubiquitous Хорошо). There is also generally less palatalisation in Ukrainian – the Russian е is often realised as і in Ukrainian (eg Rus Семя vs Ukr Сімя = family). You'll eventually recognise a couple of dozen very common Ukrainian words that are very different in Russian enabling you to distinguish the two languages.
Sorry for a long answer – you’ve asked a hard question, and given the amount of linguistic mixing its difficult to give a simple way to distinguish the two languages. All I can suggest is that you study Russian in more detail – its inevitable you’ll pick up some Ukrainian words too but you can ask friends to stop you if you code switch by mistake.
