0.5mm, 1mm and 1.5mm at least. A full set (nine screwdrivers, from 0.6 to 3 mm) is much better. The USD 50 AF (which is a French-made one, but rebranded) is a sure bet. Bergeon makes a nice set, but a bit overpriced. Do not get the anti-magnetic ones, as these are way too soft.
Quality makes a difference. I'd get one quality pair (Dumont is a very good brand), and a couple of less expensive ones to start. An anti-magnetic steel pair (size 1 or 2; whereas a fine size 5 is for hairspring work only), and any brass one, as brass is softer than steel, so it won't leave any mark on the steel, and brass is anti-magnetic (AM is always obviously written on these).
Watch oil and oiler
I like fountain oilers, because they save oil, especially if you don't do all that many watches at a time. Dip oilers will work, but it is likely that your oil will be contaminated before you use much of it. A reasonable substitute is an insulin syringe with a blunted tip. Depends on local rules; here I can get them without a prescription.
Movement holder or movement rings
It's just barely possible to work without them, but only barely. The genuine Bergeon 4040 is a classic. Do not get the Chinese/Indian ones, their sides are not parallel.
Unless your memory is a lot better than mine. Take pictures of the movement. If your camera won't take close-ups, you can probably hold a jeweler's loupe in front of the lens to get closer. Generally best with the flash off.
Possibly not right away if you get a very high quality screwdriver set, you'll need it if you get a $15 screwdriver set, and you'll need it eventually no matter what. I'd rather have cheap screwdrivers and a sharpener than a good set without. You'll need a sharpening stone: buy a double-sided USA Norton "India" stone (orange/brown): one side is coarse, the other one is fine. Always put some PETROL on it BEFORE sharpening anything on it, or the stone will get clogged up with metal fragments. For a last fine sharpening job, you can use an Arkansas stone (they're gray or bluish/white colored). Always use oil with this stone.
If you're as nearsighted as I am, you can just barely manage without one, but they are too cheap to do without. You'll want a x4 or so to get tools between the loupe and the watch, and a x12 or so to inspect extremely close. Drill some holes in the body to reduce fogging. Good brands : Asco, Bergeon. An Optivisor (headband mounted stereo magnafyer) is nice. The real ones have a prism that helps with close vision. Cheaper ones just have the lens, you wind up having to go cross-eyed to see.
Jaxa-style is the best cheap version. If you go for the Asian copy of the Jaxa tool, remember you can still buy a good Bergeon set of replacement bits.
One of the easiest tools to substitute, but inexpensive.
For installing crystals and bezels, and for closing stubborn snap-on backs.
For installing and removing plastic crystals, the only way to access some watches with one-piece cases
For holding stems and similar.
Rubber dust blower : To remove dust, or to dry a part.
A tool to remove the strap/bracelet from the watch
I like the cheap Bergeon 6767 (the one with fine tip).
I use artist's watercolor trays from WalMart, less than $1.
Rodico cleaning putty
Putty-like thumbtack substitutes can work, but the real stuff isn't expensive. The older Green one is okay, but the Premium gray leaves no residues at all.
Basically like wrapping tissue, but pre-cut in 3-4 inch squares.
Go for the black Bergeon one, expensive, but do not take a risk with those fragile hands !
Small containers for movements
I like 2 oz condiment containers with lids from a restaurant supply store.
I started with a footstool on my desk to get close to the right level, built a custom bench later. Always clean it well before doing any watchmaking job, that's a priority.
A mechanical watch cleaner (spins the parts around in jars of cleaner) does nearly as good a job as a good ultrasonic, a better job than a cheap ultrasonic, and is far easier.
L and R watch cleaning chemicals may cost more than a used mechanical machine, but make a huge difference. Non-ammoniated will last longer for occasional use, the ammonia will evaporate. If you can't afford both wash and rinse, get the wash and substitute Naptha (lighter fluid) or paint thinner for the rinse.