Cheapest: walk everywhere.
Paris isn't a very nice place for biking (though it's improving, but slowly), but you can do it. There's a public short-term bike rental service called Vélib: you take a bike from one of the ubiquitous stands (only within the Paris city limits and in a short range outside, not in the whole suburbs) and bring it to another stand within half an hour. The yearly pass is 29€ and you can use store the Vélib pass on your Navigo card (see below).
For public transport (buses, metros, trains, trams: ticketing is unified), get a Navigo pass that covers the zones that you'll be traveling to (zone 1 is the Paris municipalities and zones 2–5 make concentric rings). During the week-end, you can travel everywhere in Île-de-France regardless of zones, so pick the zones you need during the week. Since you'll be a student, get the student pass: Imagine'R. In addition to being about half the price of a normal pass, a few places give discounts for it. You'll need to have some proof that you're a student in order to get the Carte Imagine'R, and you have to mail-order it, it's a bit of a hassle. You will need a photocopy or scan of your student card or attestation from the French university you'll be visiting. Imagine'R is valid for one year starting on the first of a month of your choice between September and December, and you have to send in your application at least 3 weeks before.
Taxi is mildly expensive and typically outside of a student's budget.