The for most central and nice areas to live are Westen, Nordend, Ostend and Sachsenhausen but are also the most expansive ones. In addition it is not so easy to get an apartment their. It might take some time but maybe you are lucky and someone is looking for a short term tenant. In addition there is a pretty new city quarter called Europa Viertel which is still relatively central. It is not as old fashioned as the other parts of the city ans still a lot of work going on but that's why you find there more apartments to rent than in the other parts of town.