I lived in Texas for many years and have loads of family and friends still there, some with DC in the school system even now. I still visit frequently, as well, so I can probably answer any specific questions you have in addition to the below. Huge post incoming!
HEAT:
It is unbelievably hot. It will surprise you. Weeks in a row over 100*F (37+c) are common in the summer. Texas has two seasons: hot and brown from April-ish to November-ish, then slightly less hot and green from December-ish to March-ish. Grass grows in the "winter" there, and it very rarely drops to freezing temperatures at all. On the off chance that a flake of snow were to fall, the entire region would roll up the sidewalks and shut down.
Most places in Texas and the American southwest have what they call "a dry heat" meaning there is very low humidity and it doesn't really feel as hot as it actually is. However, on the Gulf Coast (Corpus Christi), the humidity is very high, and it can be physically oppressive! Rain on the coast is no relief, either, as it's still hot, showers don't often last long, and as soon as they're over you're living in a sauna. It's bloody hot.
To cope, you simply stay indoors and run from shadow to shadow. Put reflective shades up in your car windows when you're not in it, or your steering wheel, seats, and seat belts will burn your hands. Dark tinted car windows are very highly recommended. Park in the shade everywhere you possibly can. Everywhere you go will be heavily air conditioned, and all houses and flats are equipped with strong cold air. You will be freezing indoors and need a cardigan, as you'll be coated in sweat after the 20 step journey from the car to the door.
SCHOOLS:
The first thing to know is that school options are awfully limited. You do not get a choice in schools in the public (state-funded, free) system; your address determines which school your DC attend. People put GREAT thought and work into which neighbourhood to live in because of the schools, as quality can vary dramatically.
While home-shopping, look throughout the entire area surrounding the city, and try to find the suburb and neighbourhood with the best schools that is still a convenient commute for your DH to work. As a rule of thumb, the further you live from a city center, and the more expensive your neighbourhood (as schools are funded by property tax on houses), the better your schools will be.
The second thing to know is that education over there is not on a par with British education. It simply isn't.
How school works:
Pre-school is optional (and considered more to be childminding than education) until formal school begins about age 5. They start at Kindergarten about 5, then go to First Grade the next year, Second Grade and so on, each year. The first several years are called Elementary School - kindergarten through 5th grade. Then Middle School - 6th - 8th Grade. Then High School 9th - 12th Grade. After 12th Grade, regardless of age, they graduate, and there is no further legal obligation to attend school. Grades 1 - 12 are 100% compulsory, and in most places, but not everywhere, Kindergarten is compulsory as well.
The school year is broken into 2 semesters. The year begins in August, and the first semester ends at Christmas break. The second semester begins in January, and the year ends in May. There are very few holiday days throughout the school year: a week (IIRC) for Thanksgiving in November, two weeks for Christmas, a "spring break" of one week around Easter, and the odd day here and there. Spring breaks are often staggered between schools and districts.
HERE is the calendar for Corpus Christi school district. Most schools will have the same basic calendar, with only the start and end of term (semester) dates and the dates of spring break varying.
Depending on their birthdays, your 11yo will most likely go into 6th Grade, 7yo into 2nd grade, and your 3yo will not start formal school for a couple years. I believe the rule is that if your DC is 5 on or before the day the year starts, they will go into kindergarten, but I am not 100% certain if those are the actual dates, because I know some children have been allowed to start earlier or later due to date of birth.
When you go to register your DC into school, they should be able to tell you everything you need to know, and they will give you a list of supplies DC will need. Uniforms are still quite rare in public schools in America (common in private schools), but are becoming popular in some places, so most schools just have a dress code such as no saggy trousers, all shorts and skirts to be a certain length, no bare midriff, basic things like that. The DC will need a stack of specific supplies like paper, folders, pencils, things like that, which will vary from school to school and year to year, though in public school, all books are supplied.
Food - generally only lunch, but breakfast may be an option - is paid for by you, on a schedule of per week or per month, something like that, or by using a top-up card. Families in financial hardship can receive reduced or free lunches. There are usually limited (and frankly crap) menu options, but food brought from home should always be welcome, I believe.
Options:
There are options outside the public school system, but they can be quite limited. Private schools are available everywhere, but they are not as plentiful as they are in the UK, they can be quite far away, and I don't think they generally offer transport (see driving below). I'd venture to say most are religion-oriented (Christian, largely), and they can be quite expensive. However, a private education is going to be higher quality. There are very few alternatives like Montessori schools (privately funded) and Magnet schools (state funded, usually with a focus like maths, science, art). Private schools are NOT determined by your address; you can choose any private school. Some may require students to pass entry exams, but that is not common.
After graduating High School, is University - eye-wateringly expensive, a lifetime of saving. Only with scholarships (awards for extremely excellent performance, whether academically or in sport) can education be reduced or free, and those are rather rare. Most people pay with loans. There are no other options like apprenticeship, work experience, etc., and there are very few trade schools, almost all of which are for-profit and many are even a bit scammy.
Texas is actually rather low in terms of education standards in America. Last I looked, it was near the bottom in student performance. School curriculum is determined almost exclusively by regular (I'm not sure which years, unfortunately - annually?) standardised tests - that is, teachers teach to these tests, and they have very little freedom to teach otherwise, particularly in public schools. I believe most states are similar. It's unfortunate, the parents hate it, the teachers hate it, everyone hates it, but there it is.
OTHER THINGS:
Driving:
You will have to drive. You and your DH will both need separate cars. Absolutely everyone drives in Texas (and most of America afaik) as there are really no other options. There is almost no public transport to speak of even in the major cities in Texas, and what exists is usually laughably limited and still requires driving. The public bus system is dangerous, frankly.
Schools generally run their own buses, but many do not, and many will only bus your DC in if they live far enough from the school. You will likely need to drive your DC to school.
Everything is very spread out and compartmentalised - this area (of several square miles) is exclusively residential property, this is retail/business property, this is industrial property - so you will be lucky to live anywhere that has anything in walking distance. You will just be forced to drive everywhere. The motorways are huuuuuuge and fast and can be confusing and complicated and scary, but they are necessary, and you get used to them quickly.
Miscellaneous:
- The thunderstorms are massive, loud, violent, terrifying, and absolutely beautiful. I've never experienced anything like them anywhere else. They can be really scary, but they are utterly awe-inspiring.
- The food is WONDERFUL. Blue Bell ice cream. Mexican Coca Cola with real sugar. Tex-Mex food of any kind. BBQ from a dilapidated shack. Whataburger fast food!! You may get fat!
- The people are very friendly. It's not uncommon to strike up a conversation waiting for the till at the supermarket, and in fact I know more than one happy couple who met that way. You'll hear a lot of please and thank you, excuse me, sir and maam, and niceties. People are happy to help if you're in trouble. You're expected to smile and nod back when you make eye contact with somebody. Unfortunately these things are becoming rarer in the cities, but they are still common.
- Corpus is near the border with Mexico, so there will be a lot of Spanish spoken. Obviously the main language is English, but learning a bit of Spanish can help you here and there. It is also a "spring break" destination and can get quite crowded with drunk uni students in the spring and summer holidays. If you live in a suburb, however, you'll be unlikely to encounter much of it. They tend to stick to the beaches. DO. NOT. venture anywhere near the beach during spring break. It's a massive thing there, and you'll be stuck in the worst crowds and traffic you've ever seen!
- I never witnessed much of what @DoubleNegative mentions (aside from people privately carrying guns, I assume, but I never actually saw a gun), but I lived mainly in large cities that voted for Clinton in the last election. There are some new laws that forbid guns in certain places, and any business can ban them; they are also forbidden anywhere there is alcohol served, IIRC. Spanking, I believe, is still fairly common, but I never once witnessed anybody hit a child with a belt or anything like that, and everyone I know looks down upon it rather sternly. Most of that sounds more like country life, from what I remember.
Though it's true Texas does politically value personal liberty and freedom of speech over most other rights, so they would never do anything like outlaw certain hateful words or force a business to cater to anyone. Restaurants still had signs posted stating they reserve the right to refuse service to anybody for any reason, however those were generally to discourage drunkenness and eject rowdy patrons. A person would still make the news for refusing to serve somebody over their religion or gender or orientation or race.
- It is UTTERLY MASSIVE - you simply cannot imagine how large it is until you've driven across it - but it is not the cowboy ranch that's portrayed on tv. The major cities in Texas are all pretty liberal now and perfectly modern, but you will still find the stereotypes out in the country. It is also beautiful, and there is a little bit of everything there: skiing in the panhandle, desert and dinosaur bones and mountains in the west, the massive flat great plains in the north, piney woods in the east, huge scrubby hills and cliffs in the center, gorgeous beaches on the coast. Austin is especially fun; its own little place, very proud of being weird and genuinely the live music capital of the world. I highly recommend making a weekend of it at least once; it's "only" a 3 hour drive (which is, by Texas standards, a quick trip!).
- There are a lot of misconceptions about Texas. If you're looking for boots and hats and men on horses and rodeos and cattle roping, you'll be disappointed. Not everybody drives a truck, not everybody has a twangy accent, almost nobody listens to country music, there are no six-shooters visible on belts or shootouts with the sheriff, and it's impossible to find a tumbleweed anywhere. You will still see horses in fields and cattle and other livestock out in the country, but the cities are just like any other city in America. However, you will see Texas flags and other patriotic regalia everywhere. Everywhere.
That's everything I can think of at the moment. Please do ask any specific questions you have! As I said, I spent many, many years in Texas and still visit frequently, and I've traveled and lived all over the state, so I hope I can be of some help! And I apologise for any typos or nonsense - this was a long post!!