You can find housing offers in the Kleinanziegen (classifieds section) and then look for Immobilienteil (property) which has two sections Vermietungen (flats to rent) and Wohnangebote (flats to buy). You will find these sections normally in the Wednesday and/or weekend editions. From my experience it is best to get the weekend edition of Weser Report or Weser Kurier and look for the Immobilien section in it. The second part of the Immobilien section is Vermietung and it is divided by the different areas in Bremen, so depending on your preference for the location of your apartment you can find offers easily. There are also other newspapers which have many advertisements as well.
Another possibility is to look at specialized German publications which contain only ads.
Since reading ads might seem confusing in the beginning I have compiled a small glossary which might help you to decipher them
When and ad gives a phone number you should call as soon as you can to make an appointment. Sometimes you might get preference if you are one of the first ones to call. If there are times specified call during those times, otherwise do not call before 8 am or after 9 pm. Sometimes as ad shows a Chiffre (code) instead of a phone number. In this case you should send a letter to the newspaper quoting the Chiffre number and then they will forward it to the advertiser. Although this is more work, it might pay off since you will be competing with less people.
Generally ads are placed by three types of people: landlords, real estate agents or tenants looking for someone to take over they lease before their notice period expires. If someone is leaving the apartment and is looking for a successor they will normally include something like Nachmieter gesucht in the ad and this is a good way of avoiding the agency fees. However, they might ask you to purchase some of their furniture and kitchen equipment (an Abstand). If priced reasonably this might be better than buying everything on your own. You should be careful since sometimes vacating tenants might want to make extra cash by pricing their furniture and if you don't agree to pay their price they might not forward your application to the landlord although they are not legally entitled to choose you as Nachmieter.
If you see the words Immobilien, RDM, or just Makler in an advertisement it means that the ad is by a real estate agent. In this case in addition to the rent, utilities and deposit, there will be a commission fee.
You can also place your own ad saying that you are looking for an apartment, but most of the time this is just waste of money and time.
On some Internet sites you can place an ad like this for free so maybe you will be lucky and it doesn't hurt to try.
You can find a lot of ads which are daily updated on official Bremen city website http://www.bremen.de, in the section Schwarzes Brett. Here you can also find general information on Bremen and the area.
Other also very useful websites are the following:
More links and websites can be found on the Online Housing Resources page