I'm a current grad student at Rutgers, and have lived in Highland Park for two years. I would say it's not very feasible to bike from Edison to New Brunswick; there aren't really any bike paths to speak of, and it doesn't seem very safe. People do bike from Highland Park to NB though. Again, there aren't bike lanes, but the side streets in HP are all pretty sleepy, and people bike on the sidewalks on the busier streets (when going over the bridge into NB, for example). There are places to lock bikes up around Highland Park and around Rutgers' campus as well. If you live in HP, it is also very feasible to walk to New Brunswick, and there is a NJTransit bus that runs through Highland Park to New Brunswick for days when the weather is bad. If your department is on one of the campuses other than College Ave, Rutgers has a good, free bus system connecting all of the other campuses to College Ave.
As far as grocery stores go, the Highland Park Stop n Shop isn't great, but it's not outrageously expensive. It's where I've done almost all of my grocery shopping while here, since I don't have a car. If you're into cooking, you won't be especially impressed by it, but it gets the job done, and is in a very convenient location. HP also has a nice farmers' market that runs from June - November. There are lots of different produce stalls, and a baker and handcrafted cheese stall, so you can get most of your food there when it's around.
If you have a roommate, you can expect to pay a lot less: I pay $650/month in a two bedroom with a roommate (so $1300 total). $900-1100 sounds about right for a one bedroom. Gas bills in the winter can get quite high - my first winter I had some pretty expensive ones (in the $300-400 range) before I bought some space heaters, bundled up, and got used to being a bit colder so the heating wouldn't be so much. If you'd like to be warm and cozy in the winter, I'd recommend finding somewhere that has heating included. I know several of the apartment buildings (Highland Montgomery, Harper House, and probably the other properties managed by the same groups) include heating in rent.
Hope that helps!