​1) the best way to be introduced to Warsaw is to join a free tour around the Old Town in Warsaw that is organized every day. It is free of charge but they expect to be tipped at the end of the walk. They are cool:-) You will find more details on the tours here:  
http://freewalkingtour.com/old-town-warsaw-free-walking-tour,city,2.html 
They do also the Jewish part of the city and communism theme too. If you start with them they will mention a lot of things to see and you can decide what next.

2) Royal Castle at the Castle Square  https://www.zamek-krolewski.pl/en

3) There is The Great National Theatre very close to the Old Town, at Plac Teatralny 3. You can visit their website for repertoire and a map with their exact location here: 
http://www.narodowy.pl/en.html

4) Summertime Every day  at 9:00 pm there is a fountains show free of charge.
The fountains are between the Old Town (from New Town Square) and Vistula River and the show is very impressive:
http://fontanna-multimedialna.pl/warszawa/
http://www.estrada.com.pl/1435_multimedialny_park_fontann?lang=en 

5) Warsaw Uprising Museum,  http://www.1944.pl/en/

6) Lazienki Park (Royal Park) with its Summer Palace on Water,  http://www.lazienki-krolewskie.pl/en?langset=true

7) Palace of Culture ,  http://www.pkin.pl/en/ ; English tours are organized daily from 11:30 am till 3:30 pm

8) Copernicus Science Museum,  http://www.kopernik.org.pl/en/   Great place to go with KIDS or without:-)!

9) Garden on the University Library (this one is close to Copernicus museum),   http://www.buw.uw.edu.pl/en/index.php?option=com_content&task=view&id=286&Itemid=91
Jewish Museum,  http://www.polin.pl/en

10) Trampolin Park - fantastic place to go with kids (booking in advance needed, especially in weekends)  http://www.hangar646.pl/rezerwuj/

and many more:-) 

It is also worth going to Wilanów which is a small village on the Warsaw suburbs where there is the Palace of the Polish King  Jan the 3rd : 
http://www.wilanow-palac.art.pl/ 
There is also a poster musem in Wilanów which I've heard is one of the best in Europe:  
http://www.postermuseum.pl/en/ 

Lunch / Dinner:
1) "U Kucharzy" at Dluga 52, they cook fresh stuff at your table. Celebrities' place, both Polish and international. I have seen John Malkovic there:-) - elegant  http://www.gessler.pl/#!about/cjg9 
2) "PODWALE 25" at Podwale 25 with Polish live folk music in the eavenings, inexpensive, casual, busy; 
3) "U Fukiera" at the Old Town Square, run by Polish celebrity, Magda Gessler - elegant  http://ufukiera.pl/ 
4) "  PORTRETOWA" Restaurant Piekarska 6, family run business with great traditional Polish cuisine  www.portretowa.com.pl
5) ZAPIECEK (polskie pierogarnie) - typical traditional Polish food with gorgeous PIEROGI (dumpings), not very expensive either. 
They are at Krakowskie Przedmieście 55, Dluga 1, Freta 18 and Swiętojańska 13 in the Old Town Area - casual; 
The first location is the latest and the biggest.   http://www.zapiecek.eu/main.html 

Breakfast:
To Lubię - ul. Freta 8 - from 9am
Shabby Chic Coffee and Wine Bar - ul.Piwna 20/26 - from 9am
Brunet Cafe. - ul.Miodowa 22 - from 7am
Warszawski Sznyt. - ul. Senatorska 2 - from 9am

As for night life:
BAR & BOOKS at Wąski Dunaj 20 https://www.barandbooks.pl

plenty of night clubs and bars at Mazowiecka street numbers between 7 and 11, 20 min walk from the Old Town along the Royal Route. 

There is a view point on the top of Palace of Culture with a panoramic view over Warsaw. 
However, there is a Marriott Hotel across the street with a view point too.
Lift to the top floor in Palace of Culture costs circa 30 zł and they don't serve drinks. 
Lift to the top floor bar in Marriott costs nothing and they give you a drink for 50 zł:-) 
The Marriott bar is opened from 6 pm .

For public transport in Warsaw (how to get from one point to another) go to (English version available): 
www.jakdojade.pl 
From spring till autumn you can rent city bikes on  www.veturilo.waw.pl
I hope you will find this little list useful:-)