In a hot water boiler system, what is the correct way to bleed the lines of air? Starting upstairs and going down or starting downstairs and working up to the second floor?

Depending upon how it is piped you may end up having to bleed both up and down stairs multiple times so I would start at the top and go form there.

What do you mean by "how it is piped"? Also should I do this with the furnace on (while hot) or off while the water is cold? Thanks.

There are many ways a boiler can be piped depending upon the contractor. Main header zone,primary loop zone,secondary loop zone, circulate pump zoning, electrical valve zoning, to just name a few. I always bleed while the boiler and pipe is hot. The reason is when the water is hot it gives up some of the oxygen/air content and if you bleed when cold you may have to go back and do it again. Consider installing some automatic float vents to help prevent this issue in the future.

If your boiler has not been looked over by a professional in a while I would consider having it done. They can clean the burners, combustion chamber, lube any motors and most important adjust the burner for the proper C/O and temp. This usually saves on the utility bill the cost of having it done and is not that expensive.