It's a terrible shame to throw away socks with small holes in them - considering how easy they are to repair, I just think it's a waste to toss them. Mind you, I'm not recommending for you to walk around with patches all over your suit socks (although, it'd be an interesting conversation starter!) But for homewear, gardening clothes, and so on, there's just no reason not to attempt repairs yourself.
Personally, I tend to get holes in my clothes from all manners of outdoors activity, and with how expensive clothing is, there is no way I'm re-purchasing something I'll tear a hole into the next time I'm in the garden.
Sewing used to be a skill universally known by men, women, and even children - soldiers would patch their own uniform in the field, and since you seldom had multiple sets, you had to keep your single one in the best condition possible. Women would often be the seamstresses of the home - particularly common in historical times; doubly so when clothing was expensive (as was the case through most of history.)
All you really need is thick thread (almost like embroidery thread), yarn, fabric, or really, anything.
As long as you can apply a patch - whether it be by weaving yarn into a new "fabric", or by stitching fabric over the hole, you're golden. Having a darning ball, or darning mushroom, is very handy, as it allows you to repair the sock in a stretched position.
I would strongly recommend prioritizing function over aesthetics - as your sewing skills improve, so will the quality of your stitches. For the time being, focus on actually making a strong, solid stitch which will keep the fabric in one piece.
I would recommend starting out with your old socks, underwear, or perhaps, even random rags you have around the house. When I first picked up sewing, my mother taught me to sew using reusable kitchen towels, and bits of cloth she had laying around in her sewing box.
You don't need a sewing box of your own - most sewing kits are more than adequate - I would advise getting a variety of thicknesses of thread, but beyond that, you don't need much - a standard assortment of needles should be just fine.