we live right on the border of lincoln square and albany park. i recommend it, especially if you have a toddler. there is so much stuff going on for young kids, a lot of it free. i especially love that we are in close walking distance to two large recreational parks (wells and horner) and are close to several playlots, too. we're actually moving out of town at the end of the month, and are very sad to leave this area.
we pay 1000 for a two bedroom that includes heat. this is a steal for the area. lincoln square, ravenswood and andersonville are all wonderful family areas, but are pricier that some of the more overlooked northside neighhorhoods. if you're looking for cheaper rent, i'd look in ravenswood gardens/manor, albany park, budlong woods/lincoln bend, and edgewater. they all have a family feel to them, and are relatively safe. we lived in pilsen for a year before moving to this area, and i had mixed feelings on it. i loved some aspects of it, but didn't feel comfortable there because i am white and felt like i was viewed by my neighbors as a gentrifier, which is a hot button issue there, as it should be.
everyone is right about bucktown and wicker park-- they are so expensive now! my first apartment as an adult was in ukrainian village, and i loved it there then. i love it there now, too, but it's out of our price range, for the most part. you should see division street these days. it's a hipster version of lincoln park. most of the cheaper apartments listed as being in ukrainian village or bucktown are really in humboldt park. i've had two apartments in humboldt park, pre-kids, and wouldn't choose to raise a family there. the violence is mostly gang-related, and doesn't involve you if you're not in a gang, but it's still not a pleasant experience to have your neighbor across the street gunned down in a drive-by. plus, i just don't think it's that pretty there-- too much concrete, not enough trees.
in the almost ten years (on and off) that i've lived here, things have gotten astronomically more expensive. i used to think that chicago was the dirt-cheap, midwestern version of new york or san francisco, but that's just not the case anymore. cost of living has gone up all around the country, but i can't help but feel that independent of that, chicago has gentrified itself right out of the price ranges of many families who live here. look up average daycare rates, the cost of a gallon of gas, and even what we pay for electricity and cooking gas. it has changed a lot since 2002, in my opinion.