There is no "good position" to start from. You need to know if the carb is under or over fuelling at idle once the choke is off. Without an exhaust sniffer you are going to need to check the spark plugs to get a clue. If you have to start from nothing then carefully turn the screw until it bottoms out and then wind it back a little - this will give a very lean idle mixture.
Tightening the screw leans out the mixture, loosening the screw enriches it. Just make small adjustments. If the screw doesn't really offer much resistance while it is turning the odds are that the rubber o-ring on the screw has rotted out allowing the screw to turn freely (even from just engine vibration) - this will need replacing.
Before you do that you need to make sure that the primary throttle plate is not closing too much. If the engine will idle with just a tiny amount of throttle applied then the plate is probably closing too much and it is a different screw (still near the base of the carb) that moves the throttle stop. It is labelled as the slow idle speed adjustment on diagrams.
If you look closely down the primary barrel you'll see a series of pinholes in the side of the barrel - these are called the progression holes, they allow a tiny amount of fuel to trickle through while the throttle is nearly closed. The venturi won't work at the sort of air speeds you get while the throttle is closed. The last part of your potential problems come from these getting blocked up - if adjusting the idle speed and mixture seems to make no difference the odds are one or more of the progression holes is blocked in which case the carb needs a really good clean. Another way to double check this is to open the top of the carb and have a look in the bottom of the float bowl - if you can see what looks like fine glitter then you really need to get it cleaned. The flakes are actually remnants of fuel tank that have corroded, fine enough to get through the filters but big enough to cause trouble. Worse still are small lumps of black - these are from the fuel pipes themselves, if there are any of these then you need to replace the pipes as they've started to fall apart internally (even if they look perfect on the outside). Either way if you can see foreign bodies (no matter how small) in the float bowl it will need cleaning out!