Hi Hamlette
Welcome to Hamburg, it is the most beautiful city in Germany that I have seen. I am now 4.5 years here.
I can recommend 2 schools which I went to and enjoyed myself very much as well as learned German of course -- one is Tandem, in Altona area and another is Sprachfix in Hoheluft area.
www.tandem-hamburg.de
www.sprachfix.de
I suggest you start learning properly from schools to get a good basic foundation. I have seen many foreigners who have lived here for years and then, suddenly, they went jobless and was forced to take courses by the Arbeitsamt and meanwhile, they have already learned some broken-German from the streets and it makes the learning very frustrating as well as the teaching very annoying, also it disturbs the class alot as they 'appear' to speak very well but lots of words are for colloquial and the grammar are just in a huge mess.
If money is of no issue, consider the above 2 schools. If money is an issue, go to Volkshochschule -- I dropped out after 2 lessons becos there were nearly 40 in a class and .. oh well .. you will see the difference!
Good luck!!