Well after almost 2 years I can finally post again...
Anyway to your question, I would suggest you look at an Ender 3, they can be had for less than $200 and can print out some pretty good detail. I haven't bought one yet myself, but I know several who people who own them.
There are better printers out there but these are a great value and a very popular starter printer.
There are resin printers capable of even better detail, but greater cost, strong smell and in general more for advanced 3D printer users.
There are lot of places where you can get ready made stuff to print, but I assume you are intrested in making your own stuff.
Two popular programs for making your own designs to print are Fusion 360 which is free to hobbyists, and Tinkercad which is also free.
I've been interested and following 3D printing for about 10 years now, and the advances in the past couple years are amazing. The Ender 3 is as entry level as it gets and the detail blows away what $1200 printers were doing 5 years ago.
A few years ago I was debating between getting a 3D printer or a mini-lathe. Because at that time the printers were still rather expensive and the print quality very basic I went down the machining rabbit hole and it is deep. I bought a small lathe and milling machine (Sherlines), and those have been joined by a larger lathe, a vertical mill and a horizontal mill, which combined add up to about 1-1/2 tons of cast iron in my basement... Like I said rabbit hole.
I'd say between a 3D printer and a CNC mill the major consideration is what materials you want to work in. If plastic works, the 3D printer will probably make you happier. It is cheaper and I think more capable of complex shapes. You will need to learn a 3D drawing program either way.
If you want to make brass or aluminum parts a CNC mill would not be a bad choice, but you are looking at a significantly higher price. The base CNC ready mill from Sherline is over $900, and you still need to supply the stepper motors and computer. A full CNC mill set up is around $2500.
You mentioned wood, so perhaps you are talking about a CNC router, those are cheaper, but still more than an entry level 3D printer. Most CNC routers can also mill aluminum.
Many people are under the impression that the various mini-mills and lathes are only good for wood and plastic, but they are quite capable of working in metal. I've made parts from steel with my Sherlines.