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Previously the best results I had hand drilling a medium to large hole was to;
-Use a Automatic Center Punch
-Start a hole using a bit that the head of a Split Point drill bit would use as a guide
-Then finish with the Split Point bit.

I had to drill about 60 fairly accurate 3/8" holes and this method worked pretty well. I did have to re-do about 6 of them though.
I realize that there lots of different methods that people feel works the best like using a Starter Bit, drilling slowly and so forth. One of the methods I do not feel that works is to gradually step up with drill bit size. It just creates more instances for the bit to walk in my opion.
When I went shopping for a drill press I thought that a well fitting spindle would make the most accurate holes. I also knew that the brands that I could afford would have production tolerances that would change from product to product. Some of the models in stores had more play in the spindle but they all had some. I have read in forums that even the high priced machine shop models had loose fitting spindles. The Sears one averaged out in the decent range and had a bigger model with better features so I went with it. Plus, I wanted one last Sears purchase for nostalgia reasons.

My first assumption with using a Drill Press was that you could not drill a bad hole with them. I was wrong. The biggest issue I have had with using the tool that never drills a bad hole is trying to accurately position the bit where you want it. I did a pretty thorough internet search and the answer is to let the piece you are drilling float (not clamp it down), and let the bit find it's own center. I tried this by spinning the chuck backwards while slowly pressing the bit into the work. Afterwards I would use a clamp to hold it into that position. What I found was that if you pressed the bit into the guide hole the play in the spindle would allow the bit to move on a angle when entering the hole. So what I did was to use this principle to align the bit with the hole WHILE the work is camped down in a double cross vice. I positioned the point of the bit so that it would work like a ramp and force the bit to angle if not aligned and adjust one of the slides on the vice till the point of the bit would start into the hole without being forced in at an angle. Then II would spin the chuck 90 degrees and do the other. Lastly I would split between the two until the bit would go into the hole without being forced in at a angle. This method has been working well for me so I figured I would share it. Best of luck to you drilling your next hole. Lol.

I use a tool with a floating head clamped into the chuck to align the pre-marked hole location to the center the hole location to the drill press. I have found that using a center punch the hole location can cause the drill bit to walk a bit if the punched hole is not straight up and down and even then any flashing on the punched hole can be a problem. I clamp the drill bit as far/deep into the drill press as possible then drill the hole. Depending on how big the hole it to be I often drill a pilot hole first. It works well and I don't get as much of the drill bit walking around this way.

You always have to plan/allow for tolerance buildup as no-one or no-thing is perfect (so far anyway). CAM driven tools can do a pretty good/accurate job but there can be some off-set.

I always use centering bits like a machinist freind I used to work with recommended... there are other (Self centering bits available as well online... look up the options. But this type worked really well for me. They are short and Thick shafted, and don't go anywhere other than where the drill itself aims it. If the drill has tolerances, these may not be able to get rid of them, but it hides most tolerance issues by putting the bit on the money the first time, then any subsequent size jumps are based on a nice start. I always cut the holes with the size bit I need the hole to be or a hair shy (depending on drill bit sizes available to me) then finish with the proper size. These bits are specifically for centiring the drill hole. Basically a starter.If drilling through Sheet metal (thin stuff mainly) these will do for a center and a first pilot hole. otherwise Thicker materials just need the centering bits to do just that... center the hole and the rest is done with the proper sized bit/s. Everybody has an angle, but it's easier to me to work with a solid mill... but a drill press with a short bit like this will generally lower the error factors, as long as you don't have to dril through 3 inches of steel.

If you're trying to drill with dead-nutz accuracy, you really need a milling machine! My Grizzly drill press gets regular work-outs and does OK and it does have the slight amount of play in the spindle bearings, but still one of the best and most used tools in my shop. Ont thing that made a BIG difference was replacing the cheap-o chuck with a good Jacobs chuck. I have 2 chucks set up with tapered arbors, a 1/2" I use 99% of the time, and a 1/4" for really small stuff. Also a big help to my aging eyes is a magnifying head-set - so you can really "see" where the drill is coming down on the work in close-up detail. Magnification also helps when locating the center punch "exactly" on the X you've carefully scribed on the part. Center drills help to align the spindle with the punch mark, vs a twist drill which can bend a little going into the punch-mark.
Jeff

For locating the Automatic punch , I used a method that I saw on a "Tubacain Utube Video" of feeling when it dropped into the scribe mark while lightly sliding across it.

Something else I did not expect was that with a drill press, the chips don't break off and the metal removed stays into a swirl unless you take off the pressure. Also, the bit seems to stay sharper longer. Maybe with a swirl being removed , the chips are not double cut,reducing the sharpness of the bit at a higher rate.