In the name of transparency I wasn't able to spend a lot of time in Warsaw as most of my trip to Poland was focused on Auschwitz-Birkenau. Something I have long wanted to get the privilege to see.
Warsaw is the capital city of Poland and it is likely you'll at the very least need to transit through Poland to get to other beautiful places around the country. As of this post, its population is around 1.8 million people and it boasts two international airports and two train stations as well as a very useful public transportation system.
Public Transportation
The public transportation system in Warsaw consists of an extensive above ground tram system, buses that go practically everywhere and a limited underground train system.
When walking near the above ground tram stations you'll notice the ticket machines near by (they could be across the street from the stop as the tram is sometimes in the middle of the road). There are several ticket options to choose from. If memory serves me right they're all done by time. 24 hours, 48 hours, and one week. Something like that.
There are also two public transportation zones in Warsaw. There should be no need to get a ticket that includes the second zone as it consists of stops that are a considerable distance outside the city limits.
Just make sure once you get your ticket, the first mode you get on, whether it be the tram, a bus, or the underground, find one of the stamp machines inside and verify your ticket. After that, there's no need to verify your ticket again on a bus, tram or train. The machine will put a time on the back side of your ticket slip that has the time of when the ticket was validated and 24 hours, 48 hours or one week later it will then become invalid based on the time it was validated. It appeared the public transportation system in Warsaw is mostly a honor based system.
I mostly used the above ground tram to get around the city as the tram is easy to use, convenient and has extensive coverage of the city. I also used the buses which combined with the trams can get you anywhere you could possibly want to go.
Must Do's
Palace of Culture and Science

You can't miss the Palace of Culture and Science in Warsaw. It's one of the tallest buildings in the city. It's also near the downtown train station so if you're passing through the city or returning by train, I can assure you, you'll see it.
It's very photogenic so if you're tight on time, at the very least stop by for a photo. If you have more time for exploring you can go inside to tour the building and it has a observation area at the top where you can get some to the best views of the city.
Old Town Market Square
Old Town Market Square just gives off Eastern European vibes. It's very much worth seeing. If Warsaw is seeing favorable weather, it's picturesque and beautiful. In the wintertime there is a rink set up here where family and tourists alike can skate around for a few minutes.
St. John's Archcathedral

In Old Town Market Square you'll find St. John's Archcathedral. This cathedral dates back to 1390. It is a UNESCO World Heritage Site.
As with everything else in Warsaw, during the German invasion and occupation, the cathedral was 90% destroyed by the Nazi's. May I suggest paying it a visit at night? It is beautifully lit up at night as you can see in the image above.
Old Town Castle Square and the Royal Castle of Warsaw
It's a great place to stop and find a bite eat, or even better, a drink. Not exactly in the Old Town Castle Square, but a few steps southeast toward the Presidential Palace you can find food and drink establishments to take a break and grab a good bite to eat, and or, a drink.
If you're admiring the square, all you need to do is turnaround and right there you will see the Royal Castle of Warsaw.
Presidential Palace

A few steps southeast from the Old Town Square you can find the Presidential Palace of Poland. Unfortunately when I visited Warsaw, it was hidden behind scaffolding. Clearly it was going under some repairs. How convenient that these things are all within a few feet of each other, right
Airports
Please pay attention to the fact that there are two international airports servicing Warsaw. They're both in considerably different areas of the city. If you're not paying attention it is possible that you could be flying into one and leaving from the other.
Lotnisko Chopina Airport

This airport is straight south of Warsaw and can be accessed by a public bus or a taxi.
Warsaw Modlin Airport

Modlin Airport is much further away from Warsaw than its counterpart. To get from Modlin to Warsaw, there's a coach shuttle bus that will drop you off across the street from Warszawa Centralna train station at the Marriott. It takes almost an hour to get from Modlin to central city Warsaw. There is also a train but it runs less frequently.