In palestra

Stasera ero in palestra.

Io sollevo i pesi in palestra. Io vado in palestra cinque giorni a settimana.

"Io sollevo i pesi" is grammatically correct, but it would not be the best way to define your activity.

You can say "Faccio sollevamento pesi" or Faccio pesistica"

if you do olympic weightlifting.

The most common expression is "fare palestra" which has a general meaning of going to the gym and implicitly implies that you do one of those activities practiced in a gym, but normally used to say that you lift weights and use gym equipments, but not at a level of olympic weightlifting, just to keep in shape or strengthen the muscles. In this case I would simply say: "Io faccio palestra cinque volte alla settimana" or "Io vado in palestra cinque volte alla settimana" 

In palestra

Stasera ero in palestra.

Io sollevo i pesi in palestra. Io vado in palestra cinque giorni a settimana.

Correct, not a single error, but ...

Slightly more /Italian like/ :

In palestra sollevo pesi.

Vado in palestra cinque giorni a settimana.

Why ?

The verbal voice implies the pronoun in the majority of cases (and surely in these two cases),

so unless you need special emphasis (like making a distinction between what YOU do and what OTHERS do),

the pronoun is simply dropped.

Or (we enjoy long sentences, but not extremely long ones) :

In palestra sollevo pesi e ci vado cinque giorni a settimana.

Here /ci/ is a substitute for: / the context = palestra / + / in /

Beware: /ci/ has several meaning, it is never too early to start using it!