First, let's dump that hand powered saw/miter box.  It gets the job done, but it takes forever.  We're still on a lean budget, but shelling out ~$110 for a 10 inch electric miter saw makes a lot of sense.  This won't be your forever saw, but it should do fine for most projects.  It can handle up to a 4x6 stock which will cover 99% of what a beginning wood worker will throw at it.  
If you’re looking to setup a shop equipped mainly with stationary power tools, I’d focus the bulk of your budget on the primary big tools. It’s easier to come up with $10 for a couple of clamps as an impulse purchase, than it is $1000 for a good table saw (TS). Most shops feature the TS, and that’s where I’d focus the bulk of my research and budget, unless you’ll primarily be using a band saw (BS). The biggies for me would be a good full size stationary table saw, planer, jointer, router and router table, and maybe a modest DC (like the HF unit for $150). With those main tools, you can build just alot using dimensional lumber or sheetgoods. A BS and DP are nice, but can be added down the road…in the meantime, a modest jigsaw and handheld power drill worked fine for early on. You’ll want a reasonable work surface, whether it’s a nice bench, or an old door. I’d add a good tape measure, squares, a chisel or two, sandpaper, and some basic clamps, then would add more clamps, block plane, and other extras as you go. (Ask family members for gift cards to Rockler, Woodcraft, Amazon, Lowes, HD, etc….). $3k is doable if you’re selective….the used market can be your friend if the right deals come along.
As an x aerospace machinist I have run manual lathes and mills and programmed and run CNC as well. IMO a manual lathe is far more versatile and useful than a CNC for the average DIY buff. A CaN C is basically a very accurate production machine but to spend time programming, setting up, proving out just for a couple or several pieces is not practical ( unless you have money and time to spare)

I could not consider a 12" jointer, given that the equipment had to be moved down the stairs to my basement, and the cost would blow my budget. What I wanted in an 8" jointer was true tables, a fence that is solid and easy to adjust, a cut depth gauge that is reliable, and long tables that aid in flat­tening longer bowed planks. I have found that the Taiwanese tools have come a long way in the past 20 years. I purchased the King KC-80FX 8-inch jointer, with lever adjust parallelo­gram tables. The system arrived in a good state of tune, and the well-written manual includes a full parts list and exploded parts diagrams. The tables were extremely heavy; more about getting things down the stairs later. The jointer is reasonably priced, it runs smoothly and it is well made.

Ok, the leap from $1,000 to $2,500 is a big one.  I certainly didn't make it at one time.  It took me years.  But I know folks that decided they wanted to get into woodworking and dropped at least $2,500 getting themselves outfitted.  When you do make the jump, the thought process becomes much less about making sure you can get the job done and becomes more about having quality tools to get the job done.
One of the great furniture woods, Mahogany has a reddish-brown to deep-red tint, a straight grain, medium texture and is moderately hard. It takes stain very well, but looks great with even just a few coats of oil on it. For an even more distinguished look, exotic African Ribbon-Striped Mahogany adds amazing grain and texture elements to this already beautiful species of wood.
Those two tools comprise the most basic power tools you need to start woodworking. In addition, you'll need some hand tools. A hammer is an obvious necessity, and can be bought cheaply. A tape measure is a must- have for marking out lengths. A ruler or straight edge is needed to turn your measurements into straight lines for cutting, and can be clamped to a work piece to use as a saw guide. Speaking of clamps, they are important for joining pieces together for gluing, screwing or nailing. Most woodworkers have a lot of them, and you'll never have as many as you need. For now, just buy a few 6-inch and 12-inch clamps and add more as you need them.
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The circular saw is pretty much the first tool I grab for any project. You can use it for both rough cutting your lumber to get started on a project or making finish cuts before final assembly of your project. You can use it to make half lap joints and a variety of other joinery methods. While a table saw or stationary miter saw might make a certain task easier, it is hard to beat the cost, portability, and versatility of a circular saw. While most of these come with a blade, here is a good all-around blade that I use.
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