Brunswick is an inner city suburb in Melbourne. It lies approx. 5km north-west of the CBD.
The two main roads from the CBD to Brunswick are Sydney Road and Lygon Street. From the City, Elizabeth St turns into Royal Parade which then turns into Sydney Road. From the City, Russel St turns into Lygon St. Although it sounds confusing, the distance traveled from the City is roughly 5km. You could probably walk to Brunswick in less than an hour and a half.
For more information see the Brunswick Transport Page on Wikipedia
Brunswick caters for all taste buds. Many of the best eateries in Brunswick reside along Sydney Road and Lygon St. I have noted some excellent restaurants for the budget conscious.
Kake Di Hatti 128 Lygon St Brunswick (03)9387 7771. Tasty South Indian Restaurant located on Lygon St. The food is excellent and excellent for your wallet. Mains start from $9 and it's BYO (There is also a small wine shop across from the restaurant which does nice local wines as well) Kake Di Hatti is closed on Mondays.
Warwick Thai 400 Sydney Rd Brunswick (03)9381 2111. Call to see if they are open, as they tend to have long periods of closure. Warwick Thai has fabulous Thai food; I recommend their Chicken Skewers. Mains start around $10.50 however keep in mind, the servings are huge. If you decide to go, don't go very hungry, as sometimes the service can be slow, especially on the weekends and holidays. BYO wine and beer.
Tibas 504 Sydney Rd Brunswick (03)9380 8425. Quality Lebanese food at very cheap prices. You can order a Shwarma plate for $10, and you will get rice, salad, bread, dips and meat. Again, this restaurant has large servings. They also have very nice Baklava and do great coffee. The front end of the restaurant is for take-aways, as they also do kebabs. The restaurant in alcohol free, however you can do take away on the food if you really need a beer. They are closed Tuesdays.
CERES (Cafe) Cnr Roberts and Stewart Streets, Brunswick East (03)9387 2609. Open Monday to Saturday: 8.30AM-5PM, Sundays: 9.30AM-5PM. 
A1 Bakery, 645 Sydney Rd Brunswick. A Sydney Road classic, this place offers all sorts of Lebanese bakery products with pizzas for as little as $2. edit