Praga is an affordable area.
Black riding the Metro is quite easy. Good option for entering is an elevator - there are no turnstiles before and after it. You can jump the turnstiles, walk in with someone who has a ticket (the turnstiles are big enough for two people), or if you're feeling brave, you can walk around and open the emergency exit door and walk through. None of them are alarmed, and when the door is open, it stays open and almost everyone in Warsaw automatically walks in that door instead of the turnstile. The trams and buses, including night buses, are also easy. The drivers and controllers are from independent companies so the driver doesn't care if you have a ticket or not.
Much of the regular buses and trams now have more ticket inspectors, as prices of tickets have been raised. Compared to other European cities tickets are still cheap and can be bought everywhere. The easiest is a single ticket for 4.40zł, about 1,10€.
Controllers are usually undercover, but can be spotted if you know what to look for: They are almost always male, operating mostly in groups of two. So if you see two guys at the bus stop talking with or looking at each other and then getting in at different doors, be alarmed. Also, they usually wear something like a hip belt to carry their gear. Every time I met them they were wearing a nice shirt with a collar under they normal clothes. So it is maybe some rule from their dress code.
The longer distance trains are impossible to blackride unless you're clever. You can buy a student ticket if you've got an ISIC card (an ID from your home university may not suffice and you'll have to pay the ticket inspector the difference).
There is a good website showing you how to get from one point to another by public transport in major Polish cities (in Polish and English).
You can buy cheap meals in places called "bar mleczny". There are many of them in Warsaw - list with the sample prices. In IKEA's after taking IKEA FAMILY loyalty card (you can make a fake one in the machine there, for free) you can drink as much nice free coffee and tea as you want and sit in the restaurant area with wi-fi. Dumpster diving is working quite well here. The best places are fruit and vegetable markets. Check more on Trashwiki.
There is free wi-fi in Warsaw's central train station (Dw. Centralna). Also in the old town, you can connect to free wi-fi called "UMWarszawa"
Busking is everywhere totally easy. No needs for a permit. In the Old Town and the Main square, a several buskers and the police is walking around. The usual tips are quit small. The biggest coins are 5ztl. ~ 1,2€. And it very rare that you get ones.
There are several large squats in Warsaw. You will find the Skłot Przychodnia on Skorupki 6. There's another one called Syrena.
Each museum in Warsaw has one day when entrance is for free (list from May 2015).
Zachęta - National Art Gallery
Centre for Contemporary Art Ujazdowski Castel in Warsaw
The Palace on the Isle (Łazienki Królewskie)
Museum of Warsaw
The Asia and Pacific Museum
Antonina Leśniewska Museum of Pharmacy
Museum of Hunting and Horsemanship
Museum of Independence
Museum of the Military Ordinariate
Museum of Wola
Museum of Pawiak Prison
Mausoleum of Battle and Martyrdom
Scouts Association Museum
Route of the Old Town Cultural Basements
The Royal Castle in Warsaw
Warsaw Rising Museum
Museum of Palace at Wilanów
Museum of the Earth of the Polish Academy of Sciences in Warsaw
State Archaeological Museum
Jewish Historical Institute