Use an oil weight of 30/40 standard weight. Heavier weight oil like 80 Gear oil invites dirt. Thinner weight oil washes dirt away. An older, rusted press may turn with great difficulty or not at all. Oil all points and let it sit for a day or two. It will turn much easier. Here is a link to an oil chart. Oil chart
On the last page of the oil chart, there are instructions about oiling and near the end, a comment with the answer to a on-going question; which direction should the flywheel go when the press is in operation? It should go counter clockwise with the top spinning away for the operator. The press is designed so the opening and closing of the bed and platen allow a dwell time for inserting paper and retrieving it.
This chart and others from C&P say there are 37 oil points. About 29 are oil holes and ten are joints, junctions and linkages. On newer C&P, there may be a cap on the main oil holes to keep out dirt & dust.
It is a good practice to start at the same point each time so a routine can be developed, allowing you to make sure you don’t miss any spots. I like to start on the left side with the large roller arm, one hole at each end (2). You should fill nearly to the top. Always have a rag to wipe excess oil so it doesn’t drip on the press or floor. Next oil the roller frames (2) near the back shaft. Also oil the hole on the main shaft (2), flywheel shaft bearing (2) and the back shaft (4). The flywheel oil holes (2) are somewhat hidden down on the sides next to the frame. They are critical, if not oiled, the bearing & shaft can wear until the flywheel won’t turn or breaks. While you are on the side, oil the holes for the back treadle shaft (2), rocker lock shaft (2) and throw off (2). Then put a drop or two on the throw off leakage, and throw off bar connection (2). Now move to the other side and do the same spots. At the back of the press, the main shaft roller frame collars need oil (2), the disc lever, roller bearing and pawl a drop or two and a drop in the roller frame where the hooks and springs move in and out (3). One other place of importance is the cam roller (1) inside the large gear cam. On C&P New Series there is a hole on the outside allowing access to this gear. On Old Series presses, you have to rotate the wheel until the roller is visible and accessible.
Here is a link to a pdf for a C&P New Series press naming all parts with photos. C & P NS Parts List