Academia vs. ginásio
The primary difference between academia and ginásio when referring to a fitness center is regional geography. While both relate to exercise, the choice of word signals whether you are speaking Brazilian Portuguese or European Portuguese.
Academia
A1In Brazilian Portuguese, academia is the standard and most common term for a place where people lift weights and take fitness classes. In Portugal, this word rarely means a gym and usually refers to an educational institution, such as a university or an academy of arts.
Eu vou para a academia três vezes por semana.
(I go to the gym three times a week.)
A mensalidade daquela academia é muito cara.
(The monthly fee for that gym is very expensive.)
Eles se conheceram enquanto malhavam na academia.
(They met while working out at the gym.)
Preciso comprar tênis novos para ir à academia.
(I need to buy new sneakers to go to the gym.)
Essa academia oferece aulas de ioga e pilates.
(This gym offers yoga and pilates classes.)
Ginásio
A1In European Portuguese (Portugal), ginásio is the correct term for a commercial fitness center or health club. However, in Brazil, ginásio almost exclusively refers to a large indoor sports arena or stadium (like a basketball court with bleachers) rather than a place for personal fitness.
O ginásio abre às sete horas da manhã.
(The gym opens at seven in the morning.)
Estou cansado porque treinei muito no ginásio hoje.
(I am tired because I trained hard at the gym today.)
Aquele ginásio tem piscina e sauna.
(That gym has a swimming pool and a sauna.)
Tu vais ao ginásio depois do trabalho?
(Are you going to the gym after work?)
O meu irmão trabalha como instrutor num ginásio em Lisboa.
(My brother works as an instructor at a gym in Lisbon.)
Summary
Use academia if you are in Brazil and ginásio if you are in Portugal. If you use academia in Portugal, people might think you are going to university, and if you use ginásio in Brazil, people will think you are going to a sports stadium.