Most banks have free accounts, in both dollars and soles. Some will have a restriction that it is only free when you have a minimum of like S/.750 or US$300 on the account.
This accounts usually are limited that you have to pay if you withdraw money from the counter. ( Using the ATM is free )
BCP has a 'Primera Cuenta BCP' which is free, in both dollars and soles, has Internet banking, a debit card and later on you can get a Lanpass credit card, which you accumulate airmiles to fly for free..
BCP is the biggest bank, so most people have an account there, so it is easier to transfer money around.
They offer 'debito automatico' for your subscription, and for example automatically pay your water and electricity bill without commission and risk to forget. Or you can pay these bills with Internet banking without commission.