Looking for an apartment or house to rent as from September.

Instead of posting here, you could insert an ad into this site's classifieds section or - even better - consult online property portals (e.g. immoscout24.de) or classifieds sites (e.g. kleinanzeigen.ebay.de).
Like most Western German cities, Hamburg has a difficult rental market. You should plan on sufficient time for your search (several weeks, during which you must be in Hamburg!), increase your budget (check the Hamburg Mietspiegel to see average rental levels) and be prepared to compromise.
Also keep in mind that, on top of the rent, you have to pay EUR100-200/month for utilities (electricity, heating, water, garbage, etc.) - the exact amount depends on your usage.

Great advice Beppi, much appreciated.

Here are 2 of the more popular online sites. Even if you don’t find something here it gives one a good idea of what is available. But I think you are probably underestimating the cost. To find a livable place with 2 or 3 bedrooms for € 500 sounds unlikely, even with having to pay additional utilities like Beppi mentioned. Listings show “warm”  or “kalt” Miete which means the rental price - with or without utilities included.

https://www.immobilienscout24.de/Suche/ … tep_searchhttps://www.immowelt.de/liste/hamburg?sort=relevanz