Most braces are intended for use with special-purpose, square-shanked bits. To be able to use conventional, round-shanked drill bits look out for a brace with universal chuck. The chuck of my brace has four jaws. I even use 1" Forstner bits with it. If the shank slips into the chuck, too much force has been applied.
I use a hand drill for boring small holes up to 3/8" (9mm). When using small bits up to 1/8" (3mm) the weight of the drill alone is sufficient to drive the bit into the wood. Just support the hand drill in its upright position without applying any force to the handle.
Brad point drill bits
You need a 5mm bit for the inlay dots as ready-made dots usually come in metric sizes. Alternatively, you can use a 3/8" bit if you manage to get hold of dots of that diameter. A 10mm bit for the tuner holes and an 11mm one for the bridge posts are also required. I recommend that you use properly-sized bits. A 10mm drill bit is a must for every guitarbuilder as this is the tuner hole diameter required for the vast majority of tuners. If you live in the United States, you probably won't find 10mm or 11mm drill bits around the corner, but I know from browsing the websites of larger woodworking and tool suppliers that they are available. Buy metric bits at Rockler.com
Buy a 3/4" and 1" (20mm and 25mm) bit for removing material from the cavities. My bits are special sawtooth bits intended for being used in drill presses, but using them in a brace is "allowed" too.