It occured to me that maybe you don't have a chuck for the lathe. Do you have one? Do you have a tool holder (probably for 3/8" tools)?
You need both.* If you don't have a chuck, look at the plate on the headstock. The 4 inner holes are probably for mounting that plate to the machine's spindle. There are probably 4, Allen head screws. The outer holes are probably for mounting various chucks using either 3 or 4 screws each. There are some important dimensions for a new chuck. One is the hole-circle diameter for the mounting holes. The other is the outside diameter and the diameter of the inner ledge/raised portion. It is probably a fairly standard size in mm. The first thing I would do is mount a 3- or 4-jaw chuck. Both can be "self-centering" or scroll. Three-jaw scroll and 4-jaw independent chucks are most common.
The other tool you need is a dial indicator to show whether something is centered in the chuck and the chuck is centered on the spindle. Don't skip on that tool. Inexpensive indicators like in the first attachment will work, but are considerably more difficult to set up than the type shown in the second attachment.
*There are situations where only a collet chuck is needed, but for a hobbyist and general purpose lathe, a regular chuck is more versatile.