Well, the first thing to do is find out what your state's rules are regarding home schooling. I homeschooled in Hawaii and Louisiana, and it was pretty easy in both of those places. My kids just had to take a national test each year so the state knew they really were learning what they should be based on their ages. But every state has its own rules, so that's Step 1.
Next, you can research curricula and choose something that best fits you and your child(ren). Some programs you do completely on your own. Others, you maintain regular contact via email or regular mail and send in lessons or tests, etc. You can also check your local area for homeschooling groups. Some homeschool groups get together for things like field trips to zoos or museums, or for sports. Many public schools will allow homeschool children to join their sports teams even though the child doesn't attend regular classes there. So, you can touch base with the public school your child would have attended had you not homeschooled to see what they offer. Some public schools systems also offer curriculum for homeschoolers, too.
Once you have more information, you'll get your materials and then you'll be ready to actually homeschool. Most homeschooling people have found that it works best to have a specific schedule (like, school is from 8 am to noon, Monday through Friday), and that "class" be held in a specific room. I used the kitchen and put up the alphabet on the walls, a chalkboard, etc., to make it practical. A schedule and set place encourage learning and you and your child can both get into "school mode" by following a schedule.
Pros:
As you said, being able to schedule it when you want to. You can school over the summer and take fall off if that works for you. Remember, though, that the neighbor kids (possibly your child's friends) will have the normal school schedule, and if you get involved in a homeschool group, again there will be more of a traditional schedule.
Generally, it didn't take me long to get through the lesson plans each day. Homeschooling only a few children allows you to progress at your child's pace. If they need to spend more time on math, you can do that.
You will spend lots of time with your kids and will experience things with them. My kids are all grown now, but I cannot count how many times when they were younger other moms would say to me that they really admired the relationship I have with my kids and wish they had it with theirs. Even now that my kids are grown, my DD20's friends have told her dozens of times that they wish they were as close with their moms as my DD is with me. You will never regret spending that time, reading with them, exploring with them, laughing with them, learning with them.
Cons:
Believe it or not, some people don't "get" it and if you take your children shopping with you during normal school hours, people act like you're all playing truant. LOL!
Some will say that kids need more "socialization" than they can get with home schooling. I don't believe this as my kids always were involved in other activities like Sunday school, soccer, karate, Boy Scouts, etc., but I'm listing it so you now some say this. If your kids don't socialize with others, I do think they can benefit from doing so.
Cost -- it can be pricey to get a curriculum. You can shop online to get used ones, or your local used bookstore may have some, too. You can also share with another family, etc. But cost can be a factor.
That's the short version and I wish you the best as you research this. I wish you the best!
-Dorothy (LadyZolt)