Standard taxi rank outside arrival hall (as everywhere: not the guys who try to chat you up inside the hall), price to city center between 10 and 20$.
If you like them, you can use Uber Pop. About 20-30% cheaper, but a bit of a hassle at the airport. Rather use them inside the city.

Restaurants: I love Warszawa Wschodnia by Mateusz Gessler. 24 hours open, Polish classics and European cuisine. See and be seen, too, but the food quality has never let me down.
Some would recommend U Kucharzy w Arsenale (beware that the old location in Europejski that you might read a lot about is closed), I don't like that too much, but it has flair.

Places to visit: Old & New town (Stare i Nowe Miasto), King's way (Krakowskie Przedmieście & Nowy Świat) are easy to do by foot, from practically every hotel in the city and a must. This is Disneyland.
Museum of Warsaw Uprising, and if you have time, the Historic Museum on Old Town Square (Rynek Starego Miasta).
If you have nice weather, a stroll around Park Łazienkowski, maybe with a Chopin concert in the open (free, every Sunday morning)

EDIT: just saw these concerts start mid May only...

I love taking visitors to Stara Praga (other side of the river), and show them the kind of places that the tourist board doesn't want them to see

Other than that, just feel the vibe

Welcome to Warsaw