The best source will be online from McMaster Carr, MSC, any trusted name brand supplier. Ebay is a maybe, but could easily get a counterfeit, up to the honesty of the seller.
I recommend online because many of the industrial tool dealers are not walk in friendly. They want to sell only to those who have an account and will do the work of selection for them. Also you will pay convenience store prices!
I would just go with black oxide, jobber length, reduced 1/2" shank with flats.
The standard clearance bit for a 5/8 bolt is 21/32. I would not try stepping it. Trying to take a very small cut will cause the bit to grab and wedge in the hole. You can take it in larger steps if you like, like start with a 1/4", then finish with the 21/32.
What are you driving the drill bit with? Assuming you are drilling mild steel?
If hand drilling, you'll need a drill with a lot of torque and slow speed, think 1/2" arm breaker style. Run it slow, all the down pressure you can give it, keep it straight as possible, use cutting oil, hang on for the ride!
If at all possible, use a drill press or a mill. Much more control, longer bit life, safer operation. A typical wood drill press will not go slow enough for steel that size. It will quickly burn up the bit.
Also, if steel, you could rent or borrow a mag base drill. They work well.
Black Oxide Drills
