There are Really Only a Few Essentials

It is really easy to get caught up in developing "the perfect classroom" and wanting to have this wonderful space full of beautiful educational items for your child to "enjoy" (That all cost a pretty penny).  There are also a lot of companies out there that want you to believe their stuff is the best, but really there are only a few items you MUST own to successfully teach your kids at home other than shelter and food that is.

1. The usual supplies you would need to send them to a mortar and brick school anyway: a few organizational items like folders and notebooks and something to write with and something to write on.  Having a white board and markers is great. You use less paper when you use it and its easy to erase, but paper and pencil works well too and then you have a record of the work your child has done.
2. A World Map and or Globe:  Once you have one hanging up in a convenient spot in your home (hopefully near where you do most of your lessons) you'll be amazed at how often it gets referenced.  The key is making sure it is convenient.  If its a map, don't roll it up and keep it somewhere - it really is important to have it hanging up and available.
3. A library card:  There simply is not a better way to have access to hundreds and hundreds of resources for you and your child.  Aside from books, libraries offer music, movies, activities and their librarians.  Wonderful resources in and of themselves that are severely under-used in most cases - they have quite an education and can offer up sound advice about reference tools (on and offline) as well as how to navigate their spaces, how to get involved in local book and writing clubs and your children's librarian can probably offer some of the best ideas about fabulous books to read with your child for almost any occasion.
4. A journal:  Whether you keep your journal on your computer, in a tablet, a special book or in a public blog online, you'll need a record of the lessons you complete with your child and sample work he or she has done for state requirements (which vary from state to state) and to help you successfully comply with school attendance regulations.
5. A computer and Internet hook-up: This provides you another window to the world.  Many of the educational videos we watch are accessed online.  A great number of resources I have used were found and accessed online.  The world language program we use is a series of videos and computer games,  There are online dictionaries, thesauri and a world of information - just make sure you know the basics about distinguishing between reliable information and that which is not and teach your kid those basics along with safety guidelines regarding Internet use.
6. A Stopwatch:  I know, this one may surprise you, but there are quite a few times in science and PE when you find a stopwatch necessary.  Also, setting a child against the clock to clean up, practice and hone a skill like addition, or finding things on that world map is a surprisingly frequent aspect of schooling.
7. Desire: A desire to learn, connect and be with your child through the great days and the fabulous days as well as the dismal ones.
8. Creativity: Especially on those dismal days, you may find thinking outside the box a huge help, but when your lessons are also "outside the box" - or at least seem so to your kids - everyone has more fun.
9. Time: It takes a lot of time to home school, research, prepare, etc. are all a part of your day in addition to the time you actually spend with your child.
10. Patience and Perseverance: Learning is a process, not a destination.  Patience and diligence through the tough parts of that process are essential.

Anything else is gravy!  I have a lot of gravy, but its still just gravy.  Note the Parents Spending Portion on the chart to the right, be heartened and remember how little the difference in result is between a higher portion of spending and a lower portion of spending.