Tung oil is derived from the nuts of trees that are native to Asia but have been cultivated in other parts of the world. This is a durable finish that has a rather quick drying time and is very moisture resistant. Tung oil penetrates the woods surface, soaking deep into the wood grain for a fine finish. Tung oil is great for exterior furniture, tabletops and countertops.
Saws (I'm not a fan of Japanese saws, but you can buy good new ones pretty cheap.  ) - 1 crosscut handsaw (can be a Stanley "Sharktooth" toolbox saw from Home Depot or Lowes $15).  1 rip handsaw (used ones can be found at Antique stores for $20 or less.  Try to find one that is fairly sharp to start with).  Dovetail saw (Crown has a decent gent's saw for around $25 or get the Lee Valley Veritas Dovetail saw for around $70, Highland Woodworking carries both).  A crosscut backsaw and small miterbox is also handy to have, I can't give you a recoomendation for inexpensive ones though.
What tools are needed to start woodworking? Well, this list can vary greatly depending on your budget. The best beginner woodworking tools are subjective but can vary largely depending on how much money you are willing or able to invest to start out in woodworking. It can also vary depending on if you are wanting to start out woodworking just as a hobby or as a side-hustle business. Most of the best beginner woodworking tools on this list are either tools I own, have owned, or at least have used in the past.
Non-Standard Miter Slots - This one is a downer.  One of the primary advantages of having a table saw is access to jigs that expand the saws functionality.  This is a major issue if you plan on buying after market jigs.  Given that we are limiting the cost of this buildout to $500, I am guessing that after market jigs are probably low on the priority list.  Your going to want jigs once you start researching what they enable you to do, my advice is to build your own - there are plenty of plans online.  
As a gardener/homesteader, and setting aside basic firewood tools (chain and bow saws, hatchet, axe, maul, sledgehammer and wedges), the most important woodworking tool I possess is the drawknife. This enables one to quickly and efficiently peel bark off of saplings and poles (which alone increases their durability outdoors, and is essential to prepare them for other uses), and is also good for rough shaping work....more accurate than a hatchet, and quicker than a chisel in many cases. I probably use my drawknife about as often as any of the basic carpenter's tools like a crosscut saw or an electric drill.

Right now you're thinking, "but the plans say to use a table saw for this." That's OK , because the great thing about a circular saw is you can turn it into a table saw. There are plenty of videos online showing how to do this, but you are essentially cutting a slot in a piece of plywood, mounting the circular saw to it, and flipping it over. With careful planning you can have a good basic table saw without spending hundreds of dollars, and it can come apart if you need to save space. With that setup you can cut long pieces of wood that you may not be able to brace properly to cut with the circular saw. You can also make more accurate cuts than by holding the circular saw. It can also be used to cut simple dados and to cut a bevel. You do need to be careful, as it won't have the blade guard covering the saw blade anymore.
Just how small?  Will you have an extra bedroom for your shop or will you be doing woodworking in your living room?  Do you have to put the projects and tools away every time you want to entertain or will you leave everything set up all the time?  A while back FWW had a video tour of a shop in Japan that was smaller than small.  I'll try to find it and get back.  Found it...  shows what can be done in a small space but this shop was not on a small budget so no help there.
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