All you really need is a little plastic thing of needles ($1), some pins ($1), thread (varies but even good cotton thread isn’t that much), scissors (where you might actually consider investing a little - do not use these scissors for anything else, and consider a rotary cutter if you really get into it), and fabric.
Fabric might seem like the pricey part of the equation, but consider how much a thrift store is going to charge you for a duvet or a pile of t-shirts! I have something like 30 t-shirts I spent maybe $5 on several months ago, and I’ve been working through that pile for a while.
You can turn a t-shirt into a pillow, a reusable bag, use the scraps to patch clothing, make dolls, quilts… The bits that get to be so small to be unusable for a scrap quilt you can use to stuff things.
It takes a lot of time compared to machine sewing, but it’s an activity that can be done while watching a tv show.
Needle and thread. Maybe a seam ripper. No kit required.
Are there different sizes or gauges of needles? Common colors, materials, etc. of threads?
If you’re mending you don’t need to overthink it. You really only need a thick one if you’re like trying to pierce through layers of jeans. Yes get color thread to match whatever you’re sewing. Get grey, black, white, maybe brown and that covers most things unless you really want to make it completely invisible.
There are, but if you get the little plastic thing of assorted needles you’ll be fine. Maybe $1 from Walmart.
I’d grab black cotton thread to start out with (“mercerized” is going to be the better stuff). For mending, it’s usually either about hiding your stitches so they aren’t seen at all, and it doesn’t matter, or picking something that matches what you’re fixing.
The thread that comes with kits is usually crappy polyester that will break if you look at it funny - it’s only really useful for “basting” (sewing something together temporarily to hold it in place while you do the more permanent sewing.) The other things in kits (pins, seam rippers, tiny scissors) are usually okay though.
Fixing buttons is a good project to practice on IMHO. Lots of clothes hide an extra button somewhere on a tag inside, but you can also get a nice plastic jar of mixed buttons at the dollar store usually.
Really, don’t overthink it. Even ugly stitches will hold stuff together if you put enough of them on.
I bought one of these once. Dumped them all out and wrapped in packaging tape. Its super small and a lifetime of needles. I break maybe one every other year.
Hey, that’s the one I bought a while back! It was actually an accidental purchase because I wanted dull needles for darning but now I end up using these needles over my old ones. They are quite nice, have good sizes, and there is at least one darning needle!