Hi Sarah,
Welcome aboard! OK, for travelling around Hannover, I would recommend the annual ticket from the public transport authority,
ÜSTRA. If you're going to be here for less than a calendar year, then you can buy the card on a monthly basis, which is slightly more expensive, or for an entire calendar year (reduced fee is automatically debited from your bank account each month).
However, if you live within easy distance of the school, cycling there may be an alternative, as the cycle path network in and around Hannover is generally of a good standard.
As to where to live, now that is a question to which there are many answers as there are town districts (Stadtteile)!
I live in the List district, which has the reputation of being populated entirely by teachers and university professors - not true though, as I'm neither of those! I would say it's more of a middle class area (gutbürgerlich), than say Nordstadt or Linden, which are more working class/student orientated.
I work near Roderbruch, and it takes around 15-20 minutes for me to cycle to the office, so it's not too bad!
Certain parts of Roderbruch do not have a salubrious reputation, but I'm sure there are nice parts!
The Medical High School (MHH) is situated in Roderbruch, so there may be a chance of finding some medical students to share an apartment with (Wohngemeiinschaft, or WG).
Kirchrode is nice, and it's not too far from Roderbruch, although Misburg and Gross-Bucholz are nearer and are also nice alternatives.
Hope I've been able to help you in some way.
Michael