Most popular is the Royal Route from the Old Town (north) to Wilanow Palace (south). The latter is not a must at the moment as it's gardens are extensively reshaped and the building itself, while cute, won't blow your mind if you're familiar to European baroque architaecture.
Old Town, the surroundings of Krakowskie Przedmiescie St., Nowy Swiat St., Ujazdowskie Av. (all make northern section of Royal Route) are the best what Warsaw has to offer if you look for a 'typical' tourist attractions. By Ujazdowskie Av. are two beautiful parks: Ujazdowski Park and Lazienki Park (Royal Baths). In latter there are open Chopin concerts every Sunday at high noon.
If you wan't to learn about Warsaw's troubled history visit Uprising Museum (Grzybowska 79 St.). It's very modern, interactive and educating. Same with Chopin Museum (Okolnik 1 St., close to Music Acadamy and Nowy Swiat St.).
By Vistula river is the University of Warsaw Library with splendid roof gardens. It's not exactly by the Royal Route but close to and worth visiting.
For an interesting example of socialist realism in architecture see Plan Konstytucji (Constitutional Square) and, obviously, city's tallest building, located in the very center of Warsaw, the Palace of Culture and Science.
For modern, dynamic part of the city go downtown, see the surroundings of Jana Pawla Av. The best view on the city you'll have from an observation deck on the top of Palace of Culture and Science.
Warsaw has many cool bars, pubs and clubs. You'll find them by Krakowskie Przedmiescie Av., Nowy Swiat St., Chmielna St.
If you're keen on how "new" Warsaw looks like, get into subway (at the moment there is only one line) and go to the very last station "Kabaty". It's a new suburb, also with fine restaurants and nearby Kabacki forest, great place for bikers.
Generally, as a Warsaw patriot, I have to warn you that the city is very... eclectic. It's an architectural mess, place of contrasts and I think that one will love it or hate it.
Also, Warsaw is changing rapidly. If you come back here in a couple of years you'll find it quite changed.
Hope you're going to have good time!