Minsk boasts of a well-developed transportation system, the only one to have metro in Belarus. Minsk public transport fee is BYR 950 which is about US 0.20. Quoting a Swedish lady it is faster and definitely much cheaper than that in Stockholm.
Also there are buses, trams and trolley buses - quite a choice to get you to your destination. Taxi drivers might be hard to deal with - they always seem to be keen to rip you off for a short drive (like charging 50 quid for an airport transfer when the price is about EUR 25-30). Make sure they have a yellow number plate and turn the meter on.
Getting from the airport you may opt for a shuttle bus - BYR 6000 (USD 1) or a taxi - USD 35-USD... Late at night some haggling with airport drivers is appropriate.
Minsk is safe as houses and many tourists agree that much safer than London. It is not surprising because the police in Belarus outnumbers the army.
Andrei