This has been discussed on here in the past, several times, so worth doing a search.
My own experience, which is FAIRLY extensive, but I'm NOT an expert:
1. A "cheapo" airbrush is OK if you just want to put a fairly wide "fan-shaped" but nice and even spray of a single colour (or a colourless varnish) over a complete - or a wide area of - an object such as a bowl (NOT that I'm into turning in any way);
2. A "decent cheapo" (like the cheapest Badger for e.g.) will happily handle your stains, etc, and comes with a pair of separate size nozzles which allow the spraying of reasonably fine lines, etc;
3. If you're really sure that you don't need the compressor for anything else, then IME you don't need a compressor with a tank. My Badger diaphragm device is almost silent and PROVIDED you fit a long flexible line from compressor to airbrush, then any "pulsing" which you'd feel through the line in a compressor without a tank just disappears - the hose (NOT the cheapo plastic one, but the neoprene/cotton braided, at least 2M long) will "soften" all the pulses you'd otherwise feel/see on the finish. BUT you definitely DO need a moisture/condensation trap in the line;
4. Only if you get into "airbrush art" (e.g. custom cars, bikes, helmets - with skulls/crossbones, sea shores, sunsets - you get the idea) do you really need a single-action or even double-action airbrush (such as my Badger 150). That's my opinion anyway. BUT they ARE very nice to use, BUT also need a LOT of practice to get best results (but even cheapo airbrushes need some practice before you start off on the real job too);
5. Cleanliness, and especially cleaning after use (even using water-based materials) is VITAL for good results. You'll know if you haven't cleaned thoroughly the 2nd time you come to use the tool!;
6. Agree with the above - an airbrush that will handle both a cup feed (side, below, or above, your choice) plus a small screw-topped glass jar(s) - get several - is well worth having. Especially as if spraying turned bowls, etc, the liquid quantities you'll likely be spraying are - in airbrush terms anyway - pretty big;
7. Be prepared to accept that unless you'll be spraying a large number of items (more than 3 or 4 at a time), you'll actually spend more time in prep (masking, possibly getting the correct viscosity colour, and altering pressure, etc), plus cleaning up afterwards, than you will actually spend doing the actual spraying!
As you can probably tell from the above, most of my stuff is Badger. That's not only because it's a good brand, but also because I bought most of my stuff in the 1970's/80's, when apart from Badger and DeVillbiss, there were little other decent quality choices available. I understand that these days there are other brands around, cheaper certainly, and I've heard (but had no experience of) just as good. One name that springs to mind is Ikaywata (or similar - check spelling if searching please). From last series of posts, there also seem to be some decently-priced package deals available too, but again, no personal experience here.
As above, some time ago several of us discussed this subject and the tools favoured at least twice. Included in those posts I uploaded a description of the various types of airbrush and compressor available to help with choice, plus some practice exercises (scanned from the Manual I got with my Badger 150). If you can't find those posts after a search give me a shout and I should be able to find them again - "eventually"!!
Good luck & HTH