Most S/U-Bahn trains, buses and trams run every 5-15 minutes during the day. M buses and trams run every half hour at night; U-Bahn trains run every 15 minutes on weekend nights, with N buses following their routes every half hour (starting from Hackescher Markt) on weekday nights.
Tickets can be used on all BVG, S-Bahn and local RE train services. Vending machines have instructions in English and accept coins, often bank notes and cards too. Berlin's AB travel zone contains nearly everything; you'll only need an ABC-ticket for Potsdam and Schoenefeld airport.
With an Einzelfahrschein ticket (AB-zone €2,80, ABC €3,40) you can travel one-way for up to two hours with unlimited transfers; it's cheaper to buy four tickets at once (Vier-Fahrten-Karte, €9). Buy a €1,70 Kurzstrecke (short distance) ticket if you want to travel up to three S/U-Bahn stops, or up to six stops by bus or tram. If you anticipate a lot of travelling, get the Tageskarte (day ticket, valid until 03:00 the next morning; €7,00) or the seven-day pass (€30,00). Groups of up to five people are best off with a Kleingruppenkarte (group day ticket, €19,90). The multi-day Berlin Welcomecard (€20 -43) is valid for transport and some attractions.
Before boarding the S- or U-Bahn, always validate your ticket by punching it in the yellow or red machines near the end of the platforms. On buses and trams, the machines are on board. Public transport uses the honour system, and there are regular checks by uniformed and plainclothes inspectors. If you are caught without a valid ticket you'll be fined €60 on the spot.