Hi, you have a good question, cause commonly you see both cases.
But, refering the examples you want to know, in the grammar's rules says that:
- Countries and other places (like states, cities) .. Brasil, América, Lisboa.. are capitalized.*
- Days, months and season.. sábado, fevereiro, verão.. aren't capitalized.**
- And concerning languages and studies, both cases are correct... inglês ou Inglês, matemática ou Matématica.
I hope this is helpful ;)