Av. Mercúrio, 586

Parque Ibirapuera

Pinacoteca do Estado de São Paulo

Museu da Imagem e do Som de São Paulo

Vila Madalena

Pinheiros

Casa das Caldeiras

Via Elevado Presidente Artur da Costa e Silva

Walking around Vila Madalena , young, bohemian, lots of bars, cafes and little shops
and Jardins, restaurants, shops. Good burger on Z Deli. Don't skip Pão de Queijo Haddock Lobo, the best cheese bread since 1968. 

Museum of Art of São Paulo Assis Chateaubriand the building is more interesting than the art inside, unless there's an interesting exhibition while you're there. walk around Paulista Ave. 

Parque Ibirapuera a bit crowded on weekends. While you're there, check out São Paulo Museum of Modern Art and Oca

I like Parque Burle Marx / Fundação Aron Birmann for its gardens too. A bit out of the way

Jardim Botânico a bit ot of the way too, nice if you're really into botanical gardens. 

Pinacoteca do Estado de São Paulo is another building that is worth checking out, nice renovation work, might have an interesting exhibition while you're there.

Centro Cultural Banco do Brasil - São Paulo another one to check out for interesting exhibitions. Beautiful building.

Museu do Futebol makes up for an interesting visit too. 

If you like outdoor art you can go to Beco do Batman

Hi Katie, my name is Katy too! You are in luck - there are tons of things to do for free in Sao Paulo! All the major art museums are free admission, and even the metro is very cheap - R$3.80 as of November 2016. (In US money, that's about one dollar).  I suggest, besides the art museums (such as MAC USP, Museum of Art of São Paulo Assis Chateaubriand on Av. Paulista and Pinacoteca do Estado de São Paulo and others), to stroll down Paulista Ave on Sunday, when it is closed to traffic. There are many sights to see there, including Casa das Rosaswhich is a Victorian style house you can explore and its rose gardens on the grounds. As I mentioned, MASP (free) is also there, as well as a good-sized park called Trianon or Siqueira Campos, which is full of native vegetation with winding paths. Av. Paulista on a nice Sunday is a great place to people watch, walk, and listen to music - you will find groups of musicians playing all sorts of music, from classical to rock to Brazilian folk music of the Northeast. There are lots of bicyclists too, but I don't know what it costs to rent a bike.Be sure to look for the murals on metal plaques all along the side of Hospital Santa Catarina. They are amazing!

I suggest you visit downtown some other day except Monday, when museums tend to be closed. Take the metro to Praça da Sé, go see the cathedral, Largo de São Bento where there is another old church, Pateo do Collegio (actually spelled Patio do Colegio), where the city was founded. There is a historical museum there on the spot where Sao Paulo began, called Museu Padre Anchieta.

Take the metro to Vila Mariana station, and from there you can walk to Parque do Ibirapuera, which contains museums, statues, sculptures, a Japanese house, two small lakes and is just fun to walk around. To get there on foot, walk to MAC USP and see their free exhibits, and next to this museum is a walkway bridge over the tangle of busy streets below, which will take you right to the entrance of the park.

Finally, I suggest also going to Beco do Batman, a neighborhood where lots of graffiti artists have created beautiful artwork on the walls, doors, windows and even sometimes the streets. It's an amazing place. It may seem isolated, but it's perfectly safe to walk around there and you may encounter other tourists there as well.

By the way, if you are 60 or over, you can ride the metro free.

In my opinion the best free thing to do in Sao Paulo is a walking city tour, starting at Avenida Paulista - Bela Vista, São Paulo - SP and goes down throuth R. Augusta continuing to downtown, you'll find Praça da República, Municipal Theatre of São Paulo, Viaduto do Chá, Pateo do Collegio, Mosteiro de São Bento de São Paulo, Largo São Francisco e Praça da Sé.

I will recommend you to take any of the tours this organization offer, free walking tour of Downtown or Ave. Paulista, great content, great guides and a great experience all around. 

Sao Paul free walking tours

http://www.saopaulofreewalkingtour.com

Walk on Paulista Ave , especially on Sunday