Have a look at the post-war 'MDM' district around Plac Konstytucji and along Marszałkowska, which is an example of the prestigious 'Stalinist' architecture of the time that popped up in capital cities from Kiev to East Berlin. It is all symmetrical lines of sturdy square buildings with high arcades in a sort of modernist-neoclassical mix with a distinct 1950s communist flavour. Look out for the large and rather impressive statues of heroic workers in recesses along Marszałkowska, as well as occasional architectural flourishes that are supposed to represent Polish themes, for example the 'Polish renaissance' style of Zamość.
This high-end 1950s stuff is quite different from the functional high-rise housing blocks of later years.
The Soviet War Memorial on Żwirki i Wigury is also worth a look for the highly-detailed classical-style frieze of heroic Red Army soldiers.
Other than that, there are some nice parks leading down towards the Vistula below the old town. When I was there I rode out to a flea market somewhere in the NW of the city, but it was rather underwhelming although the parks and cemetery nearby gave an insight into how ordinary Warsawers spend their Sundays.
I was last there 8 years ago, and there were signs of quite a good network of cycle paths being developed at that time, although the main arteries still meant on-road riding in heavy traffic (and with ubiquitous tram tracks!). I rode round a lot and thoroughly enjoyed it.