The Eixample area
In this area, what you are going to do will depend on how much you can afford ! The Eixample is the area where Valencians do upscale shopping. The most luxury shops are between Colón Street and the Mercado Colón. This one worths a visit, it is even more beautiful than the Central Market, with a modernist style decorated with ceramic. Inside, there are bars, restaurants and cafés.
As a student, it is rare to splash out in this area. Nevertheless, a little walk is worthy. Streets abound with modern and spacious houses. In order to see typical Spanish culture, you can go to the Museo Taurino, and see for free, torreros photos and a lot of different bullfighting objects.
For the shops, Abanicos Carbonell in Castellón Street sells wonderful Spanish fans. A souvenir from Spain, which price starts with 1,50 euros and can stretch to 6000 euros for the antique ones.