Cheap + Efficient = Walk!
Paris "intra muros", i.e. the area inside the ring road (boulevard périphérique, postal codes starting with 75) is relatively small. It has a surface of 105 square kilometers. On the North-South axis the largest distance is 10 kilometers and on the East-West 12 kilometers. These distances are rough estimates but give you a good idea of the size.
Thus if you stay in Paris "intra muros", and if you plan your sightseeing efficiently, you can do a lot by walking. If you are tired or if you want to proceed faster, use the public transport, as described in the other answers. The bicycle rental (Vélib) is another cool and interesting alternative. However, if you want it to be cheap and effective you have to you use it more than once a day. Otherwise you can use the Metro or the bus for the same price (1.70 EUR). If you are consider Vélib, note that although Paris "intra muros" is rather compact, it is far from being flat! The Vélibs are not fancy lightweight cycles but 20+ kilogram ploughs with only three speeds. They are meant to be robust.
In my opinion, public transport tickets for several days, like the "Paris visite" are hardly worth the money, unless you are heavily using the Metro. A carnet of 10 tickets (13,30 EUR) is usually more interesting.
A last word on effectiveness. If you decide to take the public transport, prefer the bus to the metro whenever feasible. Contrary to the Metro, buses are always running on the surface and you will get to see a lot of Paris that way.