Hi - new to the thread but not to the Chicago area (have been here since late 80s).
Avoid Downers Grove. It's featureless and built for cars, miles from the lake.
While the North Shore (Evanston, Highland Park, Wilmette, all along the lakeshore northwards) are all lovely, the most affordable is Evanston and it's also the closest to Chicago, with access to the city by the El making it very convenient. There are lots of rentals in Evanston, and lots of Northwestern University related activities available. Great facilities for kids, shops, schools all great.
Highland Park, Wilmette, Glencoe, etc are not really rental-land.
Public schools in the North Shore are all good.
In Chicago proper, Lakeview is nice but rents can be high for a small living space, and parking is a hassle. In general, the closer you get to the lake, the more you pay per square foot.
West Ridge, Sauganash and Forest Glen are Chicago neighbourhoods that are nice, to the NW of downtown. All diverse, MC.
In general, all the neighbourhoods on the northern edge of the city proper are nice.
Hyde Park is lovely on the south side, but a bit of an island in a sea of high crime neighbourhoods. Nice little bubble all the same, with the University of Chicago located at the heart of the community, the Museum of Science and Industry right on your doorstep, all the other museums north on Lake Shore Drive toward the south loop, and HP has a beach.
West of the City, Oak Park and River Forest are both well worth a look.
Oak Park in particular has a wide range of housing available, lots of amenities for children and teens, access by I-290 (the Eisenhower expressway) and El and Metra to the city (two El lines, one Metra), restaurants, shopping, libraries, pools, year round rink, local orchestra, chorus, and great schools. River Forest is smaller, to the west of Oak Park. There is some rental accommodation there. Great schools, esp in River Forest, and served by an El stop and two Metra stops.
Parking for renters is a hassle in Oak Park.
OP and RF are in many ways the best of both worlds, city and suburban.
No matter where you go, you will find lots for your children to get involved in, from T-ball/baseball/softball to tennis, soccer (any neighbourhood or suburb with AYSO soccer for kids is going to be family friendly and MC) art, music, maybe even a School of Rock, folk music, classes for kids at the Art Institute, swimming, skating - hockey and figure.
Your problem will be cutting down the activity list to make it manageable.
Electricity:
You can buy electric kettles in Walmart for under $20. Mine has been going strong for about five years now.
Walmart is quite a slice of life as experiences go, but they sell reputable brands of all small appliances, good range of hoovers, etc. They also do 'order online/pickup in store' that involves no shipping charges.
I would honestly buy small appliances new in the US.
Supermarkets:
The Chicago area has Aldis - I believe Chicago is Aldi Nord (or whatever the opposite of UK and Ireland Aldis, so you may note some differences but the concept is the same). Also local chains (Jewel) and several more ethnic chains (Tony's Finer Foods, for instance, Carniceria Jimenez, Pete's Fresh Market)... plus small independent local or ethnic supermarkets. Also Whole Foods Markets, Trader Joes, Targets.. and of course, Sam's Club and Costco.
I second all the reassurance about crime. Chicago is truly a tale of two or even three cities all unto itself. My suburban home is less than 15 mins on foot from a super high crime neighbourhood of the city and I often forget to lock my door, packages get left on the stoop, no graffiti, Halloween is fun, local fireworks display on the Fourth of July..
I get by in winter with a Lands End down filled coat. Even in the Polar Vortex it was fine with a few layers underneath - if it gets that cold again you won't really be out. They warn people to stay indoors. Canada Goose is probably overkill for most winter days - North Face or similar quality will see you through a winter, with scarf, hat, gloves, snowboots, insulated boots. Most of the time you will be fine without the hat, scarf, gloves, and insulated boots.