Hi,
I am also a Mom of 4, ages 9,6,3,1, and we also homeschool. We started right out from the beginning homeschooling, so we have built up gradually. We started with simply reading and the usual fun stuff with little ones - the various concepts of counting, numbers, up and down, shapes, etc. and followed up with whatever they happened to be interested in at the time. Mostly, and most basically, it was just me reading to my little one every day and helping him to sound out the letters as began to try to read himself starting with 1 and 2 letter words and moving on to include 3 letter words, etc. and explaining the various grammar and phonics rules as we went to help him sound out the words and figure out what they meant. Once he was reading well on his own, we moved on to more formal math. My oldest was reading well at 3, and has largely led his own learning, and my job has simply been to keep up and provide materials and exposure. He's like a sponge a lot of times, so that even now, he is working on Algebra, knowing all the other concepts well enough, and remembering them, to build on them well.
My second, however, prefers play over reading or study, so at 6 he is just getting to read well to begin to find it interesting enough to pursue subjects on his own. He requires a little more structure, but he is also learning well.
My 3 year old takes after my 1st, and my 1 year old is beginning to get interested in books.
So much for the progress report. Now, we are also Christian (LDS) and my children's first readers are the Book of Mormon and the Bible. We have so far been on our own, we have not been part of any group. To make sure we are meeting up to state standards, we have gotten the standards from the State of Michigan website and been able to tell that they are at least keeping up with their appropriate grades and covering all their subjects, but usually they are far ahead of the grade they would be in at their ages. Also, in homeschooling we can cover much more material and learn much more than could be done in a school setting because of the one on one time and parental supervision.
Housework: The children do their part. But, when they are as young as yours, well, it's harder, but insisting they do their parts at least by putting away their clothes and toys can go a long way to keep things within reason, and the 2 year old can do a lot with help from the older siblings, too. For example, my 1 year old puts away everyone's clothes in the proper drawers with guidance from the older ones as to which drawer, and sometimes opening the drawers for him. It helps a lot.
Ideas: You can feed the baby, cuddle with the 2 yr old at the same time, and read to the older two (well, the 2 yr old will listen and might take over the reading, but that's okay, too). If you sit on the floor leaning against the couch with the newborn in your lap, the other 3 can settle around you so they can see the book or whatever you happen to be doing. When contention breaks out, deal with it as you feel inspired, perhaps even being done with the reading session for a while or sending them all off with a book each for a time-out, etc.
There are as many ways to homeschool as their are people, but it is basically the commitment to do make sure your children learn all they need to learn, and taking personal responsibility for it. We do our homeschooling in our living room, usually, but all we do goes into the learning -- whatever trips we take, whatever activities we do, whatever place we go. For curriculum, we print worksheets off the web, and don't really have any particular company we take books from -- whatever looks like what we want at the time. For us, the cost is basically printing costs, and little else. My husband and I have many of our introductory math and science textbooks from college, and as the children are progressing, we are encouraging their use, and the children get further skills in reading and comprehension and memory retention.
Anyway, good luck and I hope everything goes well with you. If you have any questions for me, I'd be happy to respond.