Bandaging: Toys And Tools

Minimum requirements:

One pair EMT/trauma shears
You'll need them to cut away clothing to expose wounds, and they're also useful for constructing bandages, slings, etc. from ad hoc materials.

One pair Lister-style bandage scissors (5.5" or 7")
These are shaped like a small hockey stick, and have a blunt-tipped bottom blade, to slide under bandages to cut through them without cutting the patient.
Bigger ones give you more leverage, but they weigh more and take up more room in a kit, pocket, etc.

One really sharp set of pointed tissue scissors
Cleaning wounds and prepping them for bandaging may require small pieces of non-vital tissue to be trimmed away. Or in extremely horrible cases, finishing the job of amputation when a limb etc. is hanging on by naught but a shred of flesh.

One pair of tweezers/forceps
The types and varieties are endless. I have a small assortment on me every day on duty in the ER, no less than four in my movie medic kit, and a small surgery's worth in my home assortment.

Additional items you should have on hand:

Straight needles
Assorted sizes of straight needles.
For digging out small, superficial embedded splinters and other FBs (foreign bodies).
Not for doing sutures.

Safety pins, brass and stainless
For maybe $10-20, you could go to a craft or sewing store, and have dozens of these things, in sizes from micro to big enough to pin elephants' ears. And that's exactly the size range you should have, in stainless, and in brass. If you notice sets in black, they won't go amiss either. You should have, literally, about a 1# coffee can's worth of them, bagged in small ziplok craft bags, by size and material. Their utility is incredible, the size and weight negligible, and in a pinch, you're only going to have what you have.
In a local disaster, or SHTF, they will be worth their weight in gold.

Nice to have items, but not required

Hemostats, stainless and plastic, all sizes
The stainless $2 Pakistani ones work as well as the $10-20@ German ones. The cheaper Paki ones might rust sooner in frequent sterilization cycles, but that's a supply maintenance issue. Anything is better than nothing, and lifespan is a first-world problem, given the utility and cost parameters and being able to buy 10X as many for the same $$ outlay.
I use small "mosquito" sized clamps for extracting ingrown toenail portions prior to trimming the nails. (And yes, Virginia, an ingrown toenail produces a wound, in a wet, dirty, moist environment tailor-made to go septic, and untreated can become just as life-threatening as a bullet wound. Ask me how I know.) They're also good for grasping small FBs, pinning bandages in place while you're fiddling with tape rolls, clips, etc., clamping IV tubing, and a plethora of other uses. Ask the folks who work on electronics or tie trout flies how handy a third, forth, and fifth hand can be. I've accumulated enough, at this point, to do one complete open heart surgery, or run a small clinic forever, and those are just the discards from before they started recycling them, and stuff I pick up at swap meets, gun shows, etc.
For long-term storage, Break-Free/Rem-oil or WD-40 work great. Just clean it off and sterilize them before you move them into regular patient use.

For that matter, at some point you should probably get ahold of a decent surgical tech handbook, which explains how to clean, sterilize, store, and care for surgical instruments.

Tubular gauze frames
Tubular gauze, or stockinette, looks like solid woven tubes, in sizes ranging from baby fingers, up to big enough to go around a torso. The frames come in handy for fingers, toes, arms, and feet.
The gauze itself looks solid, like a knit sweater, but the key is it stretches into a fishnet stocking-configuration, and holds a sterile dressing against a wound with less supplies, incredible stretchiness, less likelihood of becoming an accidental tourniquet, while allowing more germ-killing UV near the injury, being less hot (imagine heavy wrapping in desert or near-tropical climates, pretty much like from California to Utah to Florida, or Hawaii), producing less sweat, not contributing to skin breakdown like repeated application and removal of adhesives, and providing less of a germ magnet than other wrappings, while also keeping dressings in place, and making it harder for delirious/confused/juvenile patients to pick and play with your handiwork.
It's pricey to get large rolls in every size, but it's worth it. When you can afford it, get it, one of each, and the frames, and learn how to use it. It's the medical definition of a killer ap.

You definitely don't need tubular gauze, but after you use it, you'll wonder why you ever did without it. But even if you get it, learn the old-school techniques, for the time or place when the best kit is replaced, of necessity, with the make-do type. Belt and suspenders, please.