You can open a current account (Girokonto) or a basic account (Basiskonto) at the Hamburger Sparkasse (Haspa), the Hamburger Volksbank or Commerzbank. Children, pupils, students and trainees/apprentices are usually eligible for a free account. In all other cases, account management fees are usually payable.
Some banks, for example the Consorsbank, the Deutsche Kreditbank (DKB) and the Oldenburgische Landesbank, offer free online basic accounts (Basiskonto) for anyone.
You can also compare the terms offered by other banks here: https://www.test.de/Basiskonten-im-Test-4936098-0/ (basic account (Basiskonto)) and https://girokonto.focus.de/ (current account (Girokonto)).