I've been in Sweden for 2 months now and have only eaten out maybe two times. I like to keep a tight budget and especially in Sweden it is easy to rack up the $ out of your accounts and I'd rather save money for traveling so I don't have too many recommendations on restaurants.
PLACES TO EAT:
There is a great fast food joint on a main street "Klostergatan" called Evergreen (this is the name of the place) and it can be found on the corner next to the church Sofiakyrkan which is also a bus stop for lines 1 and 3 (red and green respectively).
Other fancier more expensive restaurants are mainly located in the city center (pretty small). Everything is pretty much in one place and if you go to Juneporten (the main bus stop), the movie theater, bus transport office, restaurants, and System Bolaget (the only liquour store) can all be found here.
Here is a list of all the super markets there are in Jönköping-
--Ica (there is also one Maxi Ica which is the biggest in the town)
-- and Hemköp
The cheapest one usually is going to be Willy:s
I would recommend doing most of you shopping here, and especially if you'll live in Råslatt it'll be most convenient since you'll have one right near by. However, I have found some cheaper items at the ICA Maxi, but in general Willy:s will have the most competitive prices and it's still good quality produce.
HOW TO GROCERY SHOP
Shopping for food can be pretty self explanatory since most things you can make out what they are by how they look like, but here are a few things I needed help on:
Is this ham or turkey?
I didn't find out which one was which until a month after. Kalkon is turkey and Skinka is ham.
Most people buy here powdered detergent, which for us in the US, is new and strange.
I would recommend to get started get a box of Tvättmedel Color (can be used for both colors an whites) and also buy some Sköljmedel which is like a softener.
Other than that I think you should be well off with no problems.
Here is also map that might help in locating some things to get you acquainted just at the beginning