On a domestic flight, each ticketholder has both a checked and carryon baggage allowance. With 4 tickets, you *could* bring on 4 22" carryons and 4 "personal items" the size of a reg. handbag. I say could because I really recommend you DON'T max out your carryons, esp. when travelling w/ kids. You need your hands to manage the children, and you can't do that if your hands are full of luggage.
You should bring enough diapers for 24 hours in your diaper bag, plus 2 outfits per child, a few large ziplocs, and about a 1/2 box of wipes, just in case of airsickness. (Little kids can't hit airsickness bags, trust me on this!) Bring spare shirts for yourself and DH on board for the same reason. Also, bring enough snacks to get the kids through the equivalent of 2 meals; if you have a flight delay you will need them. Sippy cups for both kids; the airlines don't always have cups with lids. Also, you need a toy bag; you'll be in planes and airports for several hours, and the kids will need distractions. (Dad should check on power sources for that laptop if it has a DVD player!) Also, bring at least one bottle of water; one of you may need to drink to clear the ears during takeoff, and the FA's can't serve you anything until the flight is near cruising altitude.
A carseat that a child is sitting in does not normally count against your carryon allowance, but check your airline's website to be sure. The gate-checked stroller is technically supposed to be part of your checked baggage allowance, but since you are checking it in a different place than the rest of those bags, it usually does not get counted. Before you leave home, take any accessories and the hood off of your stroller and put them in your checked baggage; stuff like that on strollers can snag machinery and cause the stroller to get broken in transit. Also, fold it yourself at the end of the jetway. Don't expect the baggage handlers to know how to fold it; they don't, and they will force it and possibly break it if you leave it set up. Putting a strap around the folded stroller is a good idea, it will make sure it stays folded, even if the handlers accidentally hit the release latch.