to simplify matters:
some pubs close earlier, specially during the week. During weekends most places would be open to around 2 -3 am or even early mornig.
people arrive at parties around 23 - midnight. if you want to know what parties are held, check out:
http://www.aktivist.pl/warszawa/
or pick up a free copy of this magazine, they lie around in pubs and bars.
mazowiecka street is pretty popular with posh and well off people, there is a selection of bars and clubs, including a jazz bar. Expect door selection, but foreigners are treated better than locals.
if you want to move around Warsaw, consider either taking public transport or a taxi.
public transportation - bus/tram/ metro - single ticket (bilet normalny) costs 2.80 pln, but its worth to buy a daily ticket (bilet dzienny) for 9 pln, if you move around. valid for 24 hours, you have to stamp it in the bus/tram or at the metro station
taxis - they wait in various places in the center, prices vary. generally you pay per kilometer - from 1.60 up to 3 pln + a starting fee (6 pln). choose a taxi that has a name of company, ignore 'no names' - they'll rip you off. on the passenger side window there is always a sticker that will tell you the price for 1 km. be aware, that what is printed in big numbers is a price only for daily fare. nighttime fare is listed below.
ps. if you come by plane, ignore people who will offer you taxi in the arrival zone. just walk outside, there is taxi place operated by 2 companies. not the cheapest ones, but no scam. from the airport to the center you should pay around 30 pln, maybe some more.