“In” people just say “aller à la salle”.

“Gymnase” is quite old-fashioned unless you are talking about the place where you do competive gymnastics.

“Gym” (short for “gymnastique”) means physical exercise, but without all the apparatus found in fitness center.

I agree with you that “salle de remise en forme” seems awfully long (and awkward as well, in my opinion). However, that term does get many Google & even N-gram hits, so please take the following suggestion from a non-native speaker with all those hits in mind:
Personally, I’d use salle de sport to cover both “gym” and “fitness center” in your two English sentences [with “tous les jours” either where you have it or at the end]:

Je vais [tous les jours] à la salle de sport [tous les jours].

(link to Reverso)

Risking even further the wrath of Google, N-gram, and those who see nothing wrong with “salle de remise en forme,” I would also note that, in addition to seeing it as awkward, I see “remise/remettre en forme” as technically excluding people who go to the gym to “garder la forme” or “rester en forme,” for I see a not so subtle difference, at least in English, between “getting in shape” and “staying in shape.”

Finally (and this is not meant to question any other answer, but rather to point out the importance, as always, of context), as one who gambles much more often than I exercise, I would personally hesitate to use only “la salle” in your sentence, because those who know me well would take it to mean “la salle de jeu” and think, whether true or not, that I’m telling them that “I go to the casino every day”!