One can easily get accommodated in so called private sector: these are people who have their apartments arranged to accepts occasional tourists. It is safe, quite cheap (you can talk about how much you can pay), you get friendly company (if needed) and tips how and where. To find these kind of places just ask some more tidily looking folks on the street, or in Internet cafes and other similar places.
There is a nice and individual hostel in the center of Tirana. Tirana Backpacker Hostel, Rruga Elbasanit 85.
You can find very cheap local food markets with really good and fresh vegetables, such as tomatoes, cucumbers, green peppers, olives, etc. Local olive oil and freshly baked bread is something everyone has to taste!
Do not drink tap water in Albania.