Ukraine is relatively safe country as for European standards. Organized crime units are hardly interested in mere tourists, so the only thing that you should be wary of is fraud and disorderly conduct.
Avoiding becoming a victim of street robbery is a matter of common sense: don't flash your money around, avoid poorly lit or deserted areas, don't keep all money and documents in one place, keep an eye on your belongings, particularly in crowded places and on public transport.
As in any other country, you may meet con artists in Ukraine. Try to avoid talking to foreigners in the street, especially late at night. Hooliganism is commonly carried out by groups of teens, therefore, avoid them as far as possible.
Credit card frauds, as well as illegal currency exchange are quite widespread in Ukraine. Paying with credit card, make sure that no one sees your PIN-code; be careful withdrawing money at the ATM (it's better to use ATMs located in banks or in crowded places); never exchange currency in the street - it's illegal in Ukraine. If you see a wallet, lying on your way, don't stop and pick it up. Swindlers might have planted it in order to accuse you in stealing.
Compliance with simple precautions, based on common sense, will allow you to avoid troubles and to have a restful and pleasant visit to Ukraine.
Although, Ukrainians are considered to be hospitable and tolerant nation, there is still low tolerance to gay/lesbian couples. This is probably due to strong religious traditions in the country, or due to the fact the homosexuality was illegal until 1991. It's true that gay marriages are not legalized in Ukraine, you don't see Love Parades and there are very few places, where gay couples can be themselves without thinking twice. So, it's no fun to be a gay in Ukraine, especially in small towns with traditional tenor of life.
Gay couples are not harassed or attacked in the streets, though - nothing like this. But they might undergo irritated glances or even exclamations, if expressing their affection in public places. It should be noted, though, that display of feelings in public by heterosexual couples is often frowned upon, too. It is considered to be a scurrilous behaviour, especially by elder population.
Don't be scared away or cancel your trip to Ukraine, though. The level of acceptance may vary nationwide. Gay and lesbian travelers should avoid hand-holding and outward displays of affection unless acceptance is obvious and unmistakable. In some places, there is absolutely no tolerance whatsoever, and in others tolerance and acceptance is predicated on gays and lesbians not 'flaunting' their sexual preference.
There has been a wave of publications about racism in Ukraine. Among other things, there were statements about ruthless behaviour towards national minorities, discrimination and xenophobic public mood that develop into mortal danger for foreigners coming to Ukraine.
Being predominantly a homogeneous white community, Ukrainians are naturally somewhat reserved, when communicating with people of different nationality. This trait used to serve as strong protection during Ukraine's tough history. And it's true that there are some 'brain-washed' teenagers, who believe that "all blacks sell drugs and harass Ukrainian women." In this regard 'some' is the key word.
Statements that there are plenty of violent crimes based on racial intolerance are way too exaggerated, though. And, given that the whole country (along with the rest of the world) will be absorbed with Euro-2012, supposed racist groups will have no time to attack football fans who are going to visit Ukraine in June 2012. (No one can guarantee fans' tolerant behavior regarding their competitors, though. But these are definitely not racist motives!)