Hi, John. I am sure no expert but I tore out a couple hundred square feet of lawn yesterday and replanted. Where you are located makes some difference but it should be OK now in most of the country. The most important thing is keeping it well watered until it is established. Be sure to water 2-3 times a day even if it rains, unless you get a real gulley washer.
As far as what kind of seed, you do want a blend so problems with one variety will not wipe out your whole lawn. You can't go too far wrong with Scott's Shade blend. They want to see you have a nice lawn and know more about varieties than most homeowners. There are certainly other good blends out there and you might check the Scott's blend varieties and find another brand that is close.
The issue with lawns that I have been working on for over 20 years without finding a good answer is "How do you get your kids to cheerfully do the work?" Let us know if you find a good answer to this.