How big the pilot needs to be depends on the final as-desired hole size.
Anything less than 1/4" I go straight through.
1/4" + to 1/2" I use something around 1/8" - typically picking a less-used bit out of the index as they tend to be sharpest
1/2" > 3/4" - I use a 1/4" (or 9/32 - sharper) pilot.
To determine how big the pilot needs to be, look at the end of the final-size bit. The chisel point that is described by the intersection of the two cutting edges is the part of the bit that doesn't cut (as well) as the cutting edges. Make the pilot slightly larger than the width of the chisel and you're good to go.
Trick with drilling any hole is to listen to the drill. The cutting sound will change as the bit starts to come out through the back of the hole. With small bits, there's very little warning, especially if you're pushing hard and the cutting is moving smoothly. With larger bits, you should be moving slowly anyway, and you should hear the change in sound and know to ease off on the drilling pressure.
When drilling thicker steel that won't fit in the drill press, I first drill the pilot size in a piece of thicker scrap (3/8" or better is best), on the drill press. I clamp the scrap in place over the hole to be drilled and use that pilot hole to make sure I keep the drill bit perpendicular to the face of the material.
If the pilot hole is straight, the next drill should follow nicely.
One other thing - I have this problem on my Delta POS drill press and others would be wise to watch for the same problem. When I bear down on the drill to cut, the table gives just a bit. I've checked all of the mounting points and it's just the cheap design of the unit that causes the problem.
Anyway - when the table "gives", that changes the angle of the hole with respect to the bit. If you don't "peck" at the drill (release the pressure and reapply), then the final hole will come out at an angle.
The big risk is that when the bit punches through the back side of the material, the table will jump back into its original location, which will make the hole now no longer perpendicular to the bit and the drill will grab the part and try to lift it from the table.
My solution until I can figure out what to do about this POS is to peck-peck-peck about every 1/8" or so as I cut through. This has the advantage of breaking off any curly-ques that might be coiling off the bit as well, plus it keeps the hole from getting too far off perpendicular.
