What you should do depends on what state the hub is in. Have you greased the bearings and oiled the hub? Has someone else gone over the hub? Most of these hubs are now at least 35 years old, many over 50 years old now.
My first suggestion is that if this is a hub you have just gotten and has not been cleaned, you should flush out the hub, grease the cones/bearings, and oil through the filler with 20-weight oil. You can flush the hub by spraying WD-40 through the oiler until it runs out the sides relatively cleanly. Brown WD-40 coming out the sides is a sign of debris and gunk in the hub.
Failing that, you can flush the hub and then run on pure 20 weight oil through the oiler. The oil will gradually seep to the bearing slots, but the bearings can be run on pure oil so long as you keep oiling it regularly. It will tend to make a mess though and you have to keep the oil off the rim. I suggest using grease on the bearing seals because it helps keep your oil in the center.
If you try to just run an old Sturmey hub in as-found condition, you could have built-up gunk or debris in the hub center or in the bearings, which will lead to hub that does not run smoothly.
I would only attempt to tune and adjust the hub after cleaning. High/3d gear is with the lever fully released and the cable loose. Second is with the lever in the middle position on the shifter. First is with the lever pulled in and the cable pulled tight.