For clean and accurate wood cuts, you've gotta have the right drilling tools. Here's a rough sketch of some items to have handy in your woodworking studio as you start tackling basic projects.
The tip of the drill bit determines how accurately and cleanly it bores wood. For most woodworking jobs, the brad-point bit is just what you need. Its sharp center point helps make sure you'll place it exactly where you want it. And once it's there, that point prevents the bit from skating away as you start drilling.
Brad-point bits also score the surface wood fibers at the start of the cut, and continue doing that as the bit advances. The far left bit in the photo is the highest-quality version, with two sharp cutting lips on either side that cleanly slice wood. The second bit from the left has sharp bevels that do an okay job, but it's much less clean.
The three twist bits above are general purpose drilling tools that work better on metal than wood. They'll get by doing miscellaneous work in the shop and they do bore wood nicely along the grain. The tip of the bit in the middle, unlike the one to its right, has a better quality split point design — it'll start the cut more directly and help prevent skating. The bit on the far right has a secondary nose point, called a bullet, that also helps start a hole reliably.
In an ideal world, you'd have a set of high quality brad-points from 1⁄16″ – 1⁄2″, in 1⁄64″ increments, but that setup is pretty expensive. Instead, start with a set up to 1⁄4″, plus a 5⁄16″ and 3⁄8″, and fill in the other sizes with utility quality bits.
Forstner bits are what you'll need to bore flat-bottomed holes. The center point places the bit on the mark, then the rim enters the wood and guides the bit as it slices. You can use these to make large diameter holes up to 4″. Since a full set of quality Forstners can be very expensive, hold off on these until you need them or start with an inexpensive import set up to 2¼″.
The humble spade bit is another useful tool for drilling large holes. These are available in increments of 1/16" up to 1", and ⅛" increments beyond 1". If you're careful, you can bore a decently clean hole with these.
There are loads of specialty drill bits that you can pick up as you need them, but a good 82° countersink is essential from the start. Other countersinks to try: an old style with radially asymmetric cutting flutes, an excellent large single-flute countersink, and a small single-flute countersink that works well in wood or metal. Avoid the hardware store model — it chatters in wood and leaves a ragged surface you won't be fond of.
An electric drill-driver is probably the best way to go. You don't need a cordless model for a woodshop, but it's nice to have, especially if you'll also be using it for DIY home projects. Look for one with a lithium-ion battery if cordless is what you're after.
For larger holes and chairmaking, a traditional brace is preferred by hand-tool aficionados, and an eggbeater-style drill is great for making smaller holes.
Once you're fully addicted to woodworking, you'll also want to get a drill press. It's an investment piece so, in the meantime, a drill guide will get you by when you need to drill accurate perpendicular holes.
Since there are so many types of screw heads, a multi-tip hand driver is a better choice than accumulating tons of separate tools. In the photo above, the driver at the left is a great example of something inexpensive with a ratchet mechanism and a set of common bits stored in the handle. The model on the right is a little fancier, and has an efficient gear drive mechanism. The interchangeable bits in these sets can also be used in your electric driver.