It’s an infinitely better deal than PayPal or or international bank-to-bank transfers. You can have balances in multiple currencies. I suggest getting the debit card.
You may need an Italian bank account for bureaucratic reasons, such as if you are paying Italian taxes or have a client who wants to pay to an Italian bank. If you’re a student or here short term, you likely don’t need one and TransferWise would do it. Banks will generally require a resident permit and residency. Here are some suggested by the international community.
N26 — online bank
Banca Etica — Ethical, sustainable financing
Via dell’Agnolo 73 R, Santa Croce area
Intesa San Paolo — Piazza della Repubblica 11 R
BancoPosta — Attached to the post office, so has a lot of ATMs. You can get a regular checking account or apply for a debit card that accepts money transfer payments with an IBAN.
Branch near Piazza della Libertà: Viale Giacomo Matteotti, 33/r
Branch in the center of Florence: Via dei Sassetti, 6, 50123 Firenze FI
BNL Gruppo BNP Paribas
Via de’ Cerretani, 32R, near the Duomo