How does one get the kenmore 12102 to sew in zigzag stitches?

This video will probably be handy, although not quite your model, it should be very close. Generally, your machine should be able to do straight & zig-zag stitch without cams. Cams were usually for decorative stitches.

Kenmore 158 17550 Sewing Machine Tutorial

Parts: https://www.sewingpartsonline.com/kenmore-158-17600-sewing-machine-parts.aspx?sort.ss_cat_sort=desc&index=64

Manual: Kenmore 158 manuals 3

Depends on what stitch you intend to use. I do a lot of mending with a three step zigzag set to maximum width and minimum length -- see: http://www.picturetrail.com/sfx/album/view/24395076 A wide, short zigzag stitch is another popular choice, as is a straight stitch worked back and forth over the area: http://www.running-w-scissors.com/2014/01/tutorial-patch-mend.html

If you're just mending a seam that's come unstitched, a straight stitch or zigzag stitch similar to what's already in the seam is what you're looking for.

Make sure the stitch is set to a sewing stitch, (not a buttonhole stitch).
Set the stitch length dial to a longer stitch.
Make sure you have you feed dogs set to the "up" position.

Before you put the cam in did you make sure the machine was set on straight stitch and then select zig zag?

timing issue, needs to be serviced and re timed.

The bes way to hem a t-shirt on a household machine is to use a double needle, straight stitch. You will see two straight stitches on top and zig zag on the back. Just make sure the zig zag catched the top of the hem.

you have tread in the bottom of it. and needs a cleaning. check the tension, most set on 3

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Knobs left to right - Needle position = Center, Length to something reasonable (try 3 to start), Width - hold the knob in and turn it to other than 0.