A lot will depend on the visa type and whether you will be able to work or not. So you could be a one-income or two-income family.
I’m not in Illinois but n general I find most things here more expensive than in the UK. Groceries, insurance, utilities, mobile phone bills, eating out, clothing, furniture... petrol is cheaper but you will almost certainly drive more do it evens out. Housing can be cheaper, depends on location.
Check how it is with schooling. In most cases you can only send your children to a school in the same district that you live in. So most people choose the school first then look for a place to live in that area rather than finding a house you like and then finding there are no good schools in that district. I would push for private school to be included in the deal.
I would also find out from the employer what is being offered in terms of health care for the family members - especially if you won’t be able to/don’t want to work. The company will have to provide healthcare for him as the employee (unless it is a very small company but I find that unlikely if they are moving people around the world) but it won’t necessarily be free. I would get the full details of the plan they use and what the co-pays, deductibles and out-of-pocket maxes are. And the monthly premium. If any of you have a long-term condition such as asthma, diabetes, etc I would check that the brand of prescription drugs that works best for you is included in the plan and if not, what the cost of the medication is.
You likely will need two cars. Do you both drive already? Check the IL DMV website re the requirements for getting an IL license. Most states require you to do at least a practical test, even if you’ve been driving for years when you arrive. I also had to do a theory test as WA required this also.
Find out what the company is offering in relocation compensation - car? Rental accommodation? Trip over there for research and fact-finding (schools, housing, etc) beforehand? Flights home at Christmas? Pet relocation? Shipping costs for all your furniture or money to buy new when you arrive or furnished accommodation? Possibility to extend?