Amsterdam has fabulous public transport, even if the silly strippenkaart system is awkward to use. You have a choice of trams, buses and metro. It's clean, fast and a one-zone journey will cost you 1.60 euros. The government wants Amsterdam to put its public transport out to tender, but in a referendum a few years ago, an overwhelming majority of Amsterdammers voted to keep the GVB in public hands.
The GVB website is clear, well-written and explains everything.
For bus services, also out of town Connexxion (Dutch only)
The 92920v website is a handy Journey planner (Dutch only)
As a rule of thumb, the trams either go from the centre (central station) or traverse the city along one of the rings (3 or 10). The metro goes from central station to the suburb of Zuidoost. A second metro is currently being dug - its one of the reasons why there seems to be so much in the way of roadworks going on.
It is cheaper to buy a strippenkaart from a newsagents than a single ticket on the tram. You basically count the number of strips your journey is, usually one strip, and then fold the card over so the next strip gets stamped in the machine. Ask an Amsterdammer.
The strippenkaart system will be be phased out. The government had hoped that by 2008 eveyone will have a public transport smart card but of course the roll-out has been delayed due to a variety of problems - not least the fact that the chip in the cards was too easy to hack into. Experiments are ongoing.