I also suggest you to check Ryanair - now it's a promo, you can find RT flight for 6EUR (with cabin luggage only) OR pay extra 15EUR per 15kg of luggage (OW).
You can easy use train (WKD) S2 or S3 from/to WAW (going to/from Warszawa Centralna - Main Station) OR 175 bus (going to Krakowskie Przedmieście - Old Town BY Central Station). You can also buy 24 horus ticket for 15PLN (1EUR = 4,50PLN) - it's valid within 1st zone (all Warsaw - bus, tram, train, etc.)
SimCard - Play. You can buy in (for example) Złote Tarasy - near the Main Station in Warsaw. Cost? 1...2...3EUR (depends from included money on).