I moved a 6yo, 4yo and 2yo. The 6yo had done two years of full time school in the UK, and was going into First Grade with kids who'd done one year of half-time school.
Our schools are excellent at differentiating learning, so she was able to stay well ahead of her peers (she's now 16).
We left the UK end of June so we went straight into US school holidays, then put them into school/pre-school in Sept at the start of the next school year. You can move them partway through the year though - lots of kids do.
In terms of quitting jobs - DH had a job to come to, so he handed in his notice in the UK, took a week off, and moved to the US ahead of us. He lived in a hotel for two months.
Meanwhile I stayed back and got the house ready to sell, got everything packed for shipping, sold or gave away everything we couldn't bring with us.
The house sale was the most stressful bit. I actually borrowed some money against the value of the house, then left it in a bank account to pay the mortgage for up to a year if it didn't sell. It actually sold in Nov so it stood empty for over 4 months. I signed all the paperwork with the solicitor and estate agent before I left.
We couldn't buy a house in the US till we'd sold our UK house, so we had to rent for a while. We actually found the house we wanted to buy, and did a rent-to-buy deal with the owner, who was desperate to move out.
No idea about cats and dogs - sorry.
Before you move - get your kids vaccinated with everything they need for school in the US. If they haven't had chickenpox, get them vaccinated against it - they can be excluded from school if chickenpox appears and they aren't immune. Get copies of your GP records.
When you choose where to move, look at the school districts. Find a good school system, then look for houses in the catchment area.
Hope that all helps - good luck!