I recommend this site:
http://www.um.warszawa.pl/en/articles/warsaw-tourism and this site:
http://www.warsaw-life.com/ and this one
http://www.spottedbylocals.com/warsaw/ and Trip Advisor, plus this cool map:
http://www.use-it-warsaw.pl/
I also think it is good to visit Warsaw in May/June because of weather and flowering, but this does depend on plans and activities.
1. Old Town: includes Plac Zamkowy, the Royal Castle (if in Warsaw on Sunday, then there is free entry to the castle, which I recommend), Zygmunt's Column, Barbikan (the brick walls surrounding some of the area), and the Old Town Rynek. I would even suggest heading a bit north of Old Town into the cute New Town area (where the site of Marie Sklodowska-Curie once lived, where there is now a museum).
2. Lazienki Park, with the Chopin Monument, gardens, Orangery, white house, and palace [
http://www.lazienki-krolewskie.pl/index.php?lang=en]. There are beautiful plants, trees, and flowers around the grounds. I hear that the boat/gondola rides on the water are nice. I recommend looking out for peacocks, which are usually near the Ampitheatre. I did see a concert by the Chopin Monument, but it was not one of the Sunday Chopin classical shows offered from May until September (two shows each Sunday at 12pm and 4pm).
3. Palace of Culture and Science Observation Deck: one of the best views in Warsaw.
4. Warsaw Uprising Museum: a well-done museum with takeaway information (not just learning while there, but actual handouts that can be taken home). There is a neat aerial video of the city after the World War II and a lot of great artifacts with interviews from people involved in the conflict. This may be one of the best museums I have ever visited, which must be why it is so popular and has been updated before. Free admission on Sundays (but the video costs extra everyday).
http://www.1944.pl/en/
5. Frederic Chopin Museum: the museum is housed in a former palace. The exhibits are not only in a lovely place, but are very interactive - you can listen to Chopin's music and touch screens to learn more information. The grounds outside are lovely - there are some murals on nearby buildings that include Chopin and the famous Golden Duck statue is on the street below the museum. Be on the lookout for the black marble Chopin Benches around the city: they mark significant places to Chopin when he lived in Warsaw (with map on the bench that show current location and other sites of benches), plus they play Chopin's music when a button is pressed.
http://chopin.museum/en
6. Tomb of the Unknown Soldier
7. Krakowskie Przedmiescie and Nowy Swiat Streets: including the Presidential Palace, Warsaw University, numerous cafes/restaurants and shops, and several monuments (Copernicus and Adam Mickiewicz to name a couple). See the Palm Tree at the end of Nowy Swiat where it intersects
Jerozolimskie Avenue. If old, ornate churches are of interest, then there are some here in addition to the Old Town area.
8. Honorable Mentions (if there is extra time): Warsaw Zoo [
http://www.zoo.waw.pl/], tours of the Sejm [
http://www.sejm.gov.pl/english.html] and Presidential Palace (NOTE: these both need reservations a bit in advance - the palace in particular took a few weeks and may need special permission), Copernicus Science Center (great interactive and playful exhibits, especially for children - which is also a warning since this place has school children tours and families visiting, which may not be fun for people against lines, crowds, or rowdy children) [
http://www.kopernik.org.pl/en/], more mermaid statues (one is easy to see in the center of the Old Town Rynek, but others exist around the city, including one near the Swietokrzyski Bridge and Copernicus Science Center), Warsaw University Library (summer months have a deck on the roof open to the public with amazing city views; the surrounding gardens are also beautiful), Warsaw Ghetto area/Jewish places in Warsaw (including the remains of the ghetto wall), National Museum of Warsaw [
http://www.mnw.art.pl/index.php/pl/english_summary/], other monuments of interest (Ghetto Uprising, Warsaw Uprising, etc.), other parks/gardens (Saxon Garden, Ujazdowski Park, Skaryszewski Park), venture in Praga (beaches along the Wisla/Vistula River, see the E. Wedel factory and its store full of candy), walk along the Wisla/Vistula River (there is a great new fountain near the river to the east of the New Town area, which sometimes has light shows - certain days this includes music) and Plac Zbawiciela and Plac Konstytucji for some good food and people or architecture observation.
9. Culture: Polish cuisine, show (opera or ballet or theatre or concert), and shopping (Hala Mirowksa market (for some flower stands, meat or seafood booths, and assorted groceries - it is all an interesting experience :-P) or play it safe along Nowy Swiat and Mokotowska or at one of the city's great malls or small shops).
10. What I Wanted to See and Ran Out of Time for (still recommend): Wilanow Palace and gardens, Pole Mokotowskie (saw some of this park area ("field") since it was not too far from my school and dorm), and just about any museum I did not see (there were SO many to choose).
There is so much to see and so much to do... but these are the main hits that show so much about the city, especially if there is not too much time to really dig in deep. I further recommend this book for an interesting stay (it is written in Polish and in English):