I am looking for what equipment I need to start a minimal shop that will eventually be making nice cabinets.
I have no preference for power tools versus hand tools.
I already have a cheap contractor's table saw, a drill, and some screwdrivers.
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Hmmm. Where to start.
While a cheap contractors table saw 'can' do a lot if you're willing to work at it, a full sized table saw will be much better, and a cabinet saw would be best. If you are using sheet material you'll either need to make a special table to help you cut or have an extra pair of hands to help.
Clamps, Clamps, Clamps, you can never have enough clamps and all sizes are needed...
You will need things to smooth the wood. be that powered jointers and planers or several hand planes. Lots of sand paper, and getting Orbit sanders and belt sanders can help.
You already have a drill and screw drivers
You'll need a hammer, a rubber mallet can be (mostly) substituted with a regular hammer and a piece of scrape wood.
That will let you muddle through many projects, after that it would be more based on what projects you specifically want and how you would like to build them.
That is what I started with. I kept my eye on craigslist and garage and estate sales for a lot of my tools which worked well for me.
I am not an expert cabinet maker, but on the few that i made, I used a router a lot for dado's and edge profiles. Also a basic set of chisels and a block plane are useful and can be had fairly cheaply. Even if you get these cheap at a garage sale they can usually be brought back to life with a little elbow grease.
A joiner and planer will be very useful once you can make that leap. I got my 'yellow' planer from the blue big box store and it has served me well.
I could go on for a while, but I am at risk of being long winded.
Hope this helps.
This an iceberg of a topic - it's a difficult question to address in a short space.
Here is a downloadable PDF from Popular Woodworking magazine that introduces an occasional theme called "I Can Do That". This PDF has a list of what tools to start with and how to pick them. These are mostly hand tools (i.e., powered hand tools) but the text covers quite a bit of background and how to think about what you want to do and what you might need.
It may not answer you question completely but it's worth a read.
I am a bit late to this question, but I started out with not much more than you have. I decided that the best way to learn to build nice cabinets would be to build shop cabinets to learn on. I could work on my technique and use what I built to better organize my shop and if I made a mistake, it wasn't the end of the world.
My first recommendation is a book by Tom Clark, Practical Shop Cabinets. This is a great book that I used in building most of my cabinets. There are four tools I'd suggest you get.
I used baltic birch plywood, and was able to build my first few cabinets with just those tools. Until I got a jointer and planer, I just used dimensional hardwood from Home Depot for my face frames.