They don't cut like ordinary twist drills or holesaws at all. You need quite a bit of force, and starting them on a flat surface without a pillar drill or drill stand would be very hard.
Some thoughts:
0. Check the plug cutter is sharp. It may not be, and some careful fettling with a fine file or hone may help it cut. You do also need to clear sawdust from the hole quite often, as the design means it will tend to clog.
1. Use thick stock material to cut your plugs from, that matches the grain direction
of your chair seat. Don't forget you'll use the plug inside-out (the narrowest bit you cut is nearest the surface of the stock).
2. Measure the outside diameter of the plug cutter.
With a spade bit or a Forstner or a brace and bit (Jennings pattern or whatever)... as close to the outside diameter as you can get, but OVERSIZE... make a shallow hole in the surface of your stock.
3. That hole will act as a guide for the plug cutter as you start it freehand. You'll need to go deep, to cut past the pointy hole of the centre of the drill you used (as long as it's invisible in the finished article, that's fine). Forstners leave less of a hole, but like plug cutters, are harder to control freehand.
4. Assuming you cut the plug fine, you can break it free with either a small chisel or an electrician's screwdriver in the slot made by the plug cutter. A beter way is to make a depth mark on the outside of the plug cutter, cut as many plugs as you can to the same depth in the same piece of stock, fairly close to each other, then saw through the stock so you sever them all at the bottom of the plugs with one saw cut.
5. I've got some of the Disston set. The all-in-one countersink and pilot cutters work OK in softwood and for floorboards etc. but you'll struggle to get a clean hole for fine work.
I'd do screw recesses with a small Forstner bit, and (better) drill through some scrap clamped to the surface, although positioning will require a bit of care. As before, a pillar drill will be better.
6. I wouldn't even attempt to cut the conical ones without a pillar drill. It's got four blades that will bounce the drill around everywhere but the place you want it.
Did we mention the pillar drill? You could also use a drill stand, but s/h pillar drill's aren't expensive though - try the local free mags, as they often go very cheaply and this isn't a very demanding job.
Hope that helps a bit.
E.