The fact that you have no idea of what to call "the bit for the chain to go round" indicates that viewing a few Youtube videos is not going to do it for you.You need to study a lot more about each step in the process of working on something before you dive in, as you are approaching this neither logically or with enough caution.
As indicated above you need specialized tools to remove the freewheel/cassette, and if it is a freewheel (fairly likely with a 1986 bike) you have just made it impossible to remove it from the hub, as you have to have the rim laced tightly to the hub for leverage. You also will not be able to true/tension the wheel with a screwdriver when rebuilding - a spoke wrench is required.
As for spokes, most all shops have multiple spoke lengths in stock so there is usually no necessity to cut to length (they also need to be rethreaded if that is done).
An example of needing to be cautious is that using the taping trick only works if you are reusing the same rim again or the replacement rim that has the spoke holes next to the valve stem offset in the same manner.
Last edited by cny-bikeman; 12-24-12 at 03:10 PM.