I have an oil burning furnace with two zones - 1st flr and 2nd flr. The 1st flr is warm but the 2nd flr is cold. How do I bleed the pipes of air ?

Are you describing a hot water boiler with radiators? If so are they baseboard or stand up cast iron.

I have baseboard heaters, and as far as I can tell, there are no bleeders on them. All I can find is in the basement... above the furnace where the lines return, there are garden-hose type valves with what looks like bleeder screws next to them. The screws do not turn. I opened the valves one by one and got water out of both of them. I only did this for about 5 seconds each. Do I have to leave the valves open longer ?

Look at both ends of your baseboard radiator, hopefully one end will have a bleeder. The air is trapped in the high point of a basrboard so there should be a bleeder