As to drills, I've used HSS screw machine bits from MSC and frankly burned or broke the small sizes way too often. I now buy only Cobalt bits in the sizes down to #56 for pre-heat-treated steel, and all Titanium. I also buy solid carbide bits for use on heat treated steel (Rc of up to 61)and consider shedding a tear when I break one of those....up to $10.00 or so each, depending on size.
Get the cobalt bits from MSC...Screw Machine length is usually all the length you need. Buy them by the dozen or 10's. Also the carbide bits are good from MSC...look over the catalog and shop price.
Also do use coolant, or at least lube with TAP-EASE stick.
Also consider Carbide reamers for pivot holes, etc.
Given all the sizes you may want for knifemaking, drill bits rapidly start looking like an investment rather that minor expense.
Using the center punch, or for extreme accuracy, optical center punch, and the combined drill/countersink to start a hole is a good idea.
As to getting a short distance into steel and have it feel or act like the steel hardens, I've had this happen a number of times, even in Titanium (6AL4V). Often my problem appears that the heat of the drilling hardens the metal in th hole you are drilling or that the cutting edge of the drill bit has been dulled by the friction and/or the heat of the drilling process. When this happened and the hole had to go "exactly there", I switched to a Carbide bit and often been able to save the hole....but you are betting that doggone expensive bit on success...
Good Luck,and as my main Knifemaking Mentor says,"Speed Kills",