I do a lot of this making circuit boards so that's probably a mixed blessing here - doing hundreds of holes on a regular basis probably makes my approach a bit spendy for occasional use.
I find the boxed sets like the "microbox" in the first picture uniformly poor these days. Perhaps 20 years ago you could source useless sets for around perhaps £3 or quality for the £10-15 mark. Nowadays it doesn't matter how much you pay, if you spend more you're simply paying a bigger markup for the same rubbish. HSS just doesn't seem suited to small bits. By the sounds of it you're drilling wood so they'll probably work well enough for the immediate job but mild steel or even brass... forget it.
In any case I would say plan for breakages from the the outset, you want several spares of the right size. I'd steer you towards the bits used in electronics instead, tungsten carbide on common 1/8" shanks. The tungsten gives you a good sharp bit that wears well - handy on fibreglass boards which are hostile to cutting tools - but they are quite brittle. Second picture is how they are usually packaged, third is a variety box set from Amazon of the same thing. The likes of Amazon and eBay are fine for sourcing and likely to give best prices. Some of the "new" bits I've had in the past are clearly resharps but haven't caused any issues, after all their former life will have been in automatic machines and probably replaced to a schedule.
As for how to drive the bits, a pin vice is one option. The mini Archimedes drills used in jewellery work very well and are dirt cheap (CPC link) but the chuck won't take those common shank bits. The reverse is likely true for your drill stand - I'm not sure what you mean there but it sounds like a stand for a handheld drill not a drill press... I'm dubious but if it has no discernible runout and rises and falls perfectly vertically then perhaps.
https://cpc.farnell.com/duratool/d02331/mini-archimedes-drill/dp/TL15768?ost=Archimedes+drill
Most drills designed for this will run at a much faster RPM than you are used to - think about how cutting speed at the bit's periphery goes down with diameter at constant RPM. In the past I used a mini 12v "dentists" drill that tops out at 14,500 but nowadays use a mini bench drill that goes to 8,500. That only cost around £50 from Aim Tools a few years back but appreciate that's an outlay for a one off job. You're likely to find a Dremel a bit cumbersome in this kind of application. You don't need those higher speeds for occasional use but will need to progress slowly and carefully.
Final point, whatever you do make sure the workpiece is held down, you can't get away without simply because you are using tiny drills. Finger pressure is fine but you can't afford the work to lift as you withdraw the drill - that's a quick route to broken bits.
